Question

factor $x^2 - x - 20$. the factored expression is \boxed{}.

Explanation:

Step1: Find two numbers

We need two numbers that multiply to \(-20\) and add up to \(-1\). Let's list the factor pairs of \(20\): \(1\) and \(20\), \(2\) and \(10\), \(4\) and \(5\). Since the product is negative, one number is positive and the other is negative. We need a pair where the difference is \(1\) (because the middle term is \(-x\), so the sum of the two numbers is \(-1\)). The pair \(4\) and \(-5\) works because \(4\times(-5)=-20\) and \(4 + (-5)=-1\).

Step2: Factor the quadratic

Using the numbers we found, we can factor \(x^{2}-x - 20\) as \((x + 4)(x - 5)\). We check this by expanding: \((x + 4)(x - 5)=x^{2}-5x + 4x-20=x^{2}-x - 20\), which matches the original expression.

Answer:

\((x + 4)(x - 5)\)