Find the Missing Term in (x-4)(x+□) = x²-2x-8: Polynomial Multiplication

Examine the following problem:

$(x-4)(x+\textcolor{purple}{\boxed{?}})=x^2-2x-8$

In order to complete the missing expression on the left side, we can simply factor the expression on the right side into trinomial form (into a product of two binomials) :

$x^2-2x-8$

Proceed to factor the expression:

Note that in the given expression, the coefficient of the squared term is 1, therefore, we can (try to) factor the expression on the left side by using quick trinomial factoring:

We will look for a pair of numbers whose product equals the free term in the expression, and whose sum equals the coefficient of the first-degree term, meaning two numbers $m,\hspace{2pt}n$ that satisfy the given values:

$m\cdot n=-8\\ m+n=-2\$

From the first requirement mentioned, that is - from the multiplication, note that the product of the numbers we're looking for needs to be negative. Therefore we can conclude that the two numbers must have different signs, according to multiplication rules. Remember that the possible factors of 8 are 4 and 2 or 8 and 1, fulfilling the second requirement mentioned. This along with the fact that the signs of the numbers we're looking for are different from each other leads us to the conclusion that the only possibility for the two numbers we're looking for is:

$\begin{cases} m=-4\\ n=2 \end{cases}$

Therefore we'll factor the expression on the right side to:

$(x-4)(x+\textcolor{purple}{\boxed{?}})=x^2-2x-8 \\ \downarrow\\ (x-4)(x+\textcolor{purple}{\boxed{?}})=(x-4)(x+2)$

The missing expression is the number 2,

Meaning - the correct answer is answer B.