To factor the quadratic expression [tex]\(2x^2 - 2x - 40\)[/tex] completely, follow these steps:
1. Identify the Greatest Common Factor (GCF):
Observe that each term in [tex]\(2x^2 - 2x - 40\)[/tex] has a factor of 2. Thus, the GCF is 2. Factor out the GCF:
[tex]\[
2x^2 - 2x - 40 = 2(x^2 - x - 20)
\][/tex]
2. Factor the quadratic expression [tex]\(x^2 - x - 20\)[/tex]:
We need to find two numbers that multiply to [tex]\(-20\)[/tex] (the constant term) and add to [tex]\(-1\)[/tex] (the coefficient of the linear term, [tex]\(x\)[/tex]).
3. Find the factors:
Consider the pairs of factors of [tex]\(-20\)[/tex]:
[tex]\[
(-1, 20), (1, -20), (-2, 10), (2, -10), (-4, 5), (4, -5)
\][/tex]
We find that [tex]\((4, -5)\)[/tex] are the numbers that multiply to [tex]\(-20\)[/tex] and add to [tex]\(-1\)[/tex]:
[tex]\[
4 + (-5) = -1
\][/tex]
4. Rewrite the quadratic expression using these factors:
[tex]\[
x^2 - x - 20 = (x + 4)(x - 5)
\][/tex]
Therefore, substituting back in the expression, we have:
[tex]\[
2(x^2 - x - 20) = 2(x + 4)(x - 5)
\][/tex]
5. Verify the chosen factors are correct:
Multiply the factors to ensure correctness:
[tex]\[
(x + 4)(x - 5) = x^2 - 5x + 4x - 20 = x^2 - x - 20
\][/tex]
So, the factorization is verified.
Therefore, the completely factored form of the expression [tex]\(2x^2 - 2x - 40\)[/tex] is:
[tex]\[
2(x - 5)(x + 4)
\][/tex]
The correctly chosen answer is:
[tex]\[
2(x - 5)(x + 4)
\][/tex]
Thus, the answer is [tex]\(2(x - 5)(x + 4)\)[/tex].
