Factor completely 2x2 − 2x − 40. 2(x − 5)(x + 4) (2x − 10)(x + 4) (x − 5)(2x + 8) 2(x − 4)(x + 5)

Break Down the Problem

1. We need to factor the quadratic expression $2x^2 - 2x - 40$.
2. Identify common factors and apply factoring techniques.

Relevant Concepts

1. We can start with factoring out the greatest common factor (GCF).
2. Then we can use the method of finding factors that multiply to give the product of the leading coefficient and the constant term.

Analysis and Detail

1. Identify the GCF: The GCF of $2x^2$, $-2x$, and $-40$ is 2.
2. Factor out the GCF: $2(x^2 - x - 20)$
3. Factor the quadratic further: We need to factor $x^2 - x - 20$. We need two numbers that multiply to $-20$ and add to $-1$. These numbers are $-5$ and $4$. $x^2 - x - 20 = (x - 5)(x + 4)$
4. Combine the factors: $2(x - 5)(x + 4)$

Verify and Summarize

1. We can verify: Expand $2(x - 5)(x + 4)$: $= 2[(x^2 + 4x - 5x - 20)] = 2[x^2 - x - 20] = 2x^2 - 2x - 40$
2. The factorization is indeed correct.

Final Answer

The complete factorization of $2x^2 - 2x - 40$ is: $2(x - 5)(x + 4)$