Expand (2x+4)(x-5): Find the Missing Term in 2x²-6x+☐

Complete the following equation:

_{$(2x+4)(x-5)=2x^2-6x+\textcolor{red}{☐}$}

Simplify the given expression on the left side:

$(2x+4)(x-5)$ Open the parentheses by applying 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, the operation between terms inside of the parentheses is automatically addition. Remember that the sign preceding a term is an inseparable part of it. We'll apply the rules of sign multiplication allowing us to present any expression inside of the parentheses. The parentheses will be opened by 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:

$(2x+4)(x-5)=2x^2-6x+\textcolor{purple}{\boxed{?}} \\ \downarrow\\ \big(2x+(+4)\big)\big(x+(-5)\big)=2x^2-6x+\textcolor{purple}{\boxed{?}} \\$Let's once again write the expanded distribution law mentioned earlier:

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

And apply it to our problem:

$\big(\textcolor{red}{2x}+\textcolor{blue}{(+4)}\big)\big(x+(-5)\big)=2x^2-6x+\textcolor{purple}{\boxed{?}} \\ \downarrow\\ \textcolor{red}{2x}\cdot x +\textcolor{red}{2x}\cdot (-5)+\textcolor{blue}{(+4)}\cdot x +\textcolor{blue}{(+4)}\cdot (-5)=2x^2-6x+\textcolor{purple}{\boxed{?}} \\$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 we'll combine like terms in the expression obtained on the left side:

$\textcolor{red}{2x}\cdot x +\textcolor{red}{2x}\cdot (-5)+\textcolor{blue}{(+4)}\cdot x +\textcolor{blue}{(+4)}\cdot (-5)=2x^2-6x+\textcolor{purple}{\boxed{?}} \\ \downarrow\\ 2x^2-10x+4x-20=2x^2-6x+\textcolor{purple}{\boxed{?}} \\ 2x^2-6x-20=2x^2-6x+\textcolor{purple}{\boxed{?}}$

Therefore, according to the multiplication sign rules, the missing expression is the number $-20$,

In other words - the correct answer is C.