Expand (x+7)(x-9): Find the Missing Coefficient in x²+□x-63

Complete the following equation:

_{$(x+7)(x-9)=x^2+\textcolor{red}{☐}x-63$}

Simplify the given expression on the left side:

$(x+7)(x-9)$

Proceed to open the parentheses using the expanded distribution law:

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$

Note that in the formula template for the distribution law mentioned, we take by default that the operation between terms inside of the parentheses is addition. It's vital to remember that the sign preceding the term is an integral part of it. By applying the rules of sign multiplication we can present any expression inside of the parentheses. Open the parentheses using the above formula, first, as an expression where addition occurs between all terms (if needed),

Therefore, we'll first present each of the expressions inside of the parentheses in the multiplication on the left side as an expression containing addition:

$(x+7)(x-9)=x^2+\textcolor{purple}{\boxed{?}}x-63 \\ \downarrow\\ \big(x+(+7)\big)\big(x+(-9)\big)=x^2+\textcolor{purple}{\boxed{?}}x-63 \\$Let's write the expanded distribution law mentioned earlier:

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$

Proceed to apply it to our problem:

$\big (\textcolor{red}{x}+\textcolor{blue}{(+7)}\big)\big(x+(-9)\big)=x^2+\textcolor{purple}{\boxed{?}}x-63 \\ \downarrow\\ \textcolor{red}{x}\cdot x +\textcolor{red}{x}\cdot (-9)+\textcolor{blue}{(+7)}\cdot x +\textcolor{blue}{(+7)}\cdot (-9)=x^2+\textcolor{purple}{\boxed{?}}x-63 \\$Continue to apply the multiplication sign rules, remember that multiplying terms with identical signs yields a positive result, and multiplying terms with different signs yields a negative result. In the next step combine like terms in the expression obtained on the left side:

$\textcolor{red}{x}\cdot x +\textcolor{red}{x}\cdot (-9)+\textcolor{blue}{(+7)}\cdot x +\textcolor{blue}{(+7)}\cdot (-9)=x^2+\textcolor{purple}{\boxed{?}}x-63 \\ \downarrow\\ x^2-9x+7x-63=x^2+\textcolor{purple}{\boxed{?}}x-63 \\ x^2-2x-63=x^2+\textcolor{purple}{\boxed{?}}x-63$

Therefore the missing expression is the number $-2$,

That is - the correct answer is a'.