Table formatting: remove horizontal and vertical borders from only specific cells

Like this?

You have many possibilities, one of them is use \multicolumn command which hasn't border lines:

\documentclass{article}
\usepackage[table]{xcolor}
\newcommand\mcc[1]{\multicolumn{1}{c}{#1}} % <--- 

\setlength{\parskip}{\baselineskip}
\title{FACTORbox}
\author{Algebra Teacher Who Struggles with LaTeX Tables}
\date{January 2021}

\begin{document}

\maketitle
In the table below, I only want \textbf{borderlines around the values in black,
} not the red values above and to the left of the black colored values.

Also, I can't figure out a way to \textbf{remove the gray shading from upper left corner cell.}


\renewcommand\arraystretch{2}
\begin{tabular}{ c | c | c | }
 %  \hline
 \mcc{} & \mcc{\textcolor{red}{$x$}} & \mcc{\textcolor{red}{$+6$}} \\
    \cline{2-3}
 \textcolor{red}{$x$} & $x^2$ & $6x$ \\
    \cline{2-3}
 \textcolor{red}{$+2$} & $2x$ & $12$ \\
    \cline{2-3}
\end{tabular}


\end{document}

Edit:

Another possibilities is to use ˙blkarray package:

\documentclass{article}
\usepackage[table]{xcolor}
\usepackage{blkarray}

\begin{document}
$\renewcommand\arraystretch{2}
\begin{blockarray}{r cc}
    & \textcolor{red}{x} & \textcolor{red}{+6}   \\
    \cline{2-3}
    \begin{block}{r|c|c|}
    \cline{2-3}
\textcolor{red}{x} & x^2 & 6x \\
    \cline{2-3}
\textcolor{red}{+2} & 2x & 12 \\
    \cline{2-3}
    \end{block}
\end{blockarray}
$
\end{document}

Result is the same as before.

With the package nicematrix.

\documentclass{article}
\usepackage{nicematrix}
\usepackage{xcolor}

\begin{document}

\renewcommand\arraystretch{1.8}
$\begin{NiceMatrix}%
  [
    hvlines,
    columns-width=auto,
    first-row, code-for-first-row = \color{red},
    first-col, code-for-first-col = \color{red}
  ]
   & x   & +6 \\
x  & x^2 & 6x \\
+2 & 2x  & 12
\end{NiceMatrix}$

\end{document}

You need several compilations (because nicematrix uses PGF/Tikz nodes).