Request your help with creating generic rectangles in high school algebra worksheet

Question

I have attached a screenshot of the algebra worksheet that I am trying to create.

Format I'm trying to create:

In the code below, I have successfully (miracle of miracles) created a 2-column problem set. However, I've been stymied in my efforts to create/draw the 2 cell tables we call "generic rectangles".

If you peruse the file attachment, you'll see an example of the problem set that was originally (and badly) created in Word.

In this example, you'll notice integers colored red and blue. That's what the students write to effect the distribution of a constant located outside the parenthesis containing the binomial. So I need sufficient space around and inside the 2 cell rectangle for students to write the integers.

One last problem...

In my code, for prob #10, I can't figure out how to insert space after "(10)" so that the equation isn't bumping up against the problem number.

Thanks so much for taking time to respond to my query!

PS Because I'm a Latex newbie, there are probably superfluous packages being loaded in the preamble.

\documentclass[12pt]{exam}

%\printanswers % uncomment to print solutions

\usepackage{tasks}    
\usepackage[margin=0.5in]{geometry}

\everymath{\displaystyle}

%\unframedsolutions % uncomment to remove boxes from solutions
\setlength{\parindent}{0pt} % removes paragraph indentation

\pagestyle{headandfoot}
\header{Semester 1 Review \#3: Distribution \& Solving}{}{03/12 - 12/13/19} 

\begin{document}

\makebox[\textwidth]{Name:\enspace\hrulefill}

\vspace{4mm}

\fbox{\fbox{\parbox{7.5 in}{
\textbf {Obj. Q.8.b:} I can use the generic rectangle to distribute the constant. (10 pts. tot.) \\
\newline
Simplify by expanding with repeated multiplication.YOU WILL NOT RECEIVE CREDIT IF YOU SKIP THIS STEP!!! Your answer should only contain positive exponents.}}}

%Use the 'tasks' tool to double column itemize and enumerate
%Credit to: https://tex.stackexchange.com/questions/402222/two-column-enumerated-list
%the first item sets the spacing
\settasks{after-item-skip=8em,
          after-skip=2cm,
          label-width=3ex,
          %change '1' to 'a' if you want letters instead of numbers
          counter-format=(tsk[1]),
          column-sep=2em
          }

\begin{tasks}(2)
\task $6(1-7k)$
\begin{solution}
solution
\end{solution}
\task $5(1-10x)$
\begin{solution}
solution
\end{solution}
\task $-2(-9+5x)$
\begin{solution}
solution
\end{solution}
\task $-(7x-10)$
\begin{solution}
solution
\end{solution}
\task $-5(1+5x)$
\begin{solution}
solution
\end{solution}
\task $2(x-5)$
\begin{solution}
solution
\end{solution}
\task $4-(-5-2p)$
\begin{solution}
solution
\end{solution}
\task $9+9(-6-7n)+8k$
\begin{solution}
solution
\end{solution}
\task $-3(-2+5k)-9(10v+4)$
\begin{solution}
solution
\end{solution}
\task $10(x+7)-10(9x+1)$
\begin{solution}
solution
\end{solution}
\end{tasks}

\pagebreak

\makebox[\textwidth]{Name:\enspace\hrulefill}

\vspace{4mm}

\fbox{\fbox{\parbox{7.5 in}{
\textbf {Obj. Q.8.b:} Use the distributive property to simplify the following expression. YOU MUST USE THE GENERIC RECTANGLES PROVIDED. (10 pts. tot.) \\
\newline
Simplify by expanding with repeated multiplication.YOU WILL NOT RECEIVE CREDIT IF YOU SKIP THIS STEP!!! Your answer should only contain positive exponents.}}}
\vspace{0.5mm}

%Use the 'tasks' tool to double column itemize and enumerate
%Credit to: https://tex.stackexchange.com/questions/402222/two-column-enumerated-list
%the first item sets the spacing

\settasks{after-item-skip=8em,
          after-skip=2cm,
          label-width=3ex,
          %change '1' to 'a' if you want letters instead of numbers
          counter-format=(tsk[1]),
          column-sep=2em
          }

%[counter-format = tsk[R], label-format=\normalfont,  after-  skip=1\medskipamount](3)

\begin{tasks}(2)
\task $3x^4\cdot 4x^3$
\begin{solution}
solution
\end{solution}

\end{tasks}
\end{document}

---

This image link is shows a couple of minor formatting glitches in Bernard's VERY helpful code submitted as an answer. ]2

Answer 1

Is this what you want?

\documentclass[12pt]{exam}

%\printanswers % uncomment to print solutions.
% MACROS
%Not sure all of these packages are necessary...but the 'taks package is certainly necessary
\usepackage{amsmath}
\usepackage{cancel}
\usepackage{tikz}
\usepackage{graphicx}
\usepackage{pgfplots}
\usepackage{caption} % allows captions in minipage envir (issue w/solutions envir)
\usepackage{amssymb}
\usepackage{mathrsfs}
\usepackage{framed} %box para
\usepackage{tkz-euclide}
\usepackage{multicol}
\usepackage{tasks}
\pgfplotsset{compat=1.10}
%\displaystyle for all $$ math environments use \lim\limits for other environments

\usepackage[margin=0.5in]{geometry}

\everymath{\displaystyle}

\newcommand{\genrectangle}{\smash{\begin{tabular}{|p{20mm}|p{20mm}|}\hline\rule{0pt}{15mm} & \\\hline\end{tabular}}}

%\unframedsolutions % uncomment to remove boxes from solutions
\setlength{\parindent}{0pt} % removes paragraph indentation

\pagestyle{headandfoot}
\def\no{4}
\header{Semester 1 Review \no: Distribution \& Solving}{}{03/12 - 12/13/19}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -------------------------- DOCUMENT STARTS HERE -------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

\makebox[\textwidth]{Name:\enspace\hrulefill}


\vspace{4mm}

\fbox{\fbox{\parbox{7.5 in}{
\textbf {Obj. Q.8.b:} I can use the generic rectangle to distribute the constant. (10 pts. tot.) \\
\newline
Simplify by expanding with repeated multiplication.YOU WILL NOT RECEIVE CREDIT IF YOU SKIP THIS STEP!!! Your answer should only contain positive exponents.}}}


%Use the 'tasks' tool to double column itemize and enumerate
%Credit to: https://tex.stackexchange.com/questions/402222/two-column-enumerated-list
%the first item sets the spacing
\settasks{%
item-indent=1.5em,
after-item-skip=8em,
            before-skip=4ex,
          after-skip=2cm,
          label-width=3ex, %3ex
label-align=right,
label-offset=0.5em,
          %change '1' to 'a' if you want letters instead of numbers
         % counter-format=(tsk[1]),
         label =\arabic*),
          column-sep=2em
          }

\begin{tasks}(2)
\task $6(1-7k)$\hfill \genrectangle
\begin{solution}
solution
\end{solution}
\task $5(1-10x)$\hfill \genrectangle
\begin{solution}
solution
\end{solution}
\task $-2(-9+5x)$\hfill \genrectangle
\begin{solution}
solution
\end{solution}
\task $-(7x-10)$\hfill \genrectangle
\begin{solution}
solution
\end{solution}
\task $-5(1+5x)$\hfill \genrectangle
\begin{solution}
solution
\end{solution}
\task $2(x-5)$\hfill \genrectangle
\begin{solution}
solution
\end{solution}
\task $4-(-5-2p)$\hfill \genrectangle
\begin{solution}
solution
\end{solution}
\task $9+9(-6-7n)+8k$\hfill \genrectangle
\begin{solution}
solution
\end{solution}
\task $-3(-2+5k)-9(10v+4)$\hfill \genrectangle
\begin{solution}
solution
\end{solution}
\task $10(x+7)-10(9x+1)$\hfill \genrectangle
\begin{solution}
solution
\end{solution}
\end{tasks}

\pagebreak

\makebox[\textwidth]{Name:\enspace\hrulefill}

\vspace{4mm}

\fbox{\fbox{\parbox{7.5 in}{
\textbf {Obj. Q.8.b:} Use the distributive property to simplify the following expression. YOU MUST USE THE GENERIC RECTANGLES PROVIDED. (10 pts. tot.) \\
\newline
Simplify by expanding with repeated multiplication.YOU WILL NOT RECEIVE CREDIT IF YOU SKIP THIS STEP!!! Your answer should only contain positive exponents.}}}
\vspace{0.5mm}

%Use the 'tasks' tool to double column itemize and enumerate
%Credit to: https://tex.stackexchange.com/questions/402222/two-column-enumerated-list
%the first item sets the spacing

\settasks{after-item-skip=8em,
          after-skip=2cm,
          label-width=3ex,%
label-align = right,
label-offset = 1em,
item-indent=4em,
          %change '1' to 'a' if you want letters instead of numbers
          %counter-format=(tsk[1]),
           label =\arabic*),
          column-sep=2em
          }

%[counter-format = tsk[R], label-format=\normalfont, after- skip=1\medskipamount](3)

\begin{tasks}(2)
\task $3x^4\cdot 4x^3$
\begin{solution}
solution
\end{solution}

\end{tasks}

\end{document}