Factor completely 2x2 + 2x − 12. 2(x − 3)(x + 4) 2(x − 2)(x + 3) (2x − 4)(x + 3) (2x − 3)(x + 4)

1. Break Down the Problem

We need to factor the quadratic expression $2x^2 + 2x - 12$.

2. Relevant Concepts

To factor the expression, we can first look for the greatest common factor (GCF) and then apply factoring techniques for quadratic expressions.

3. Analysis and Detail

1. Identify the GCF: The GCF of $2x^2$, $2x$, and $-12$ is $2$.

2. Factor out the GCF: $2(x^2 + x - 6)$

3. Factor the quadratic $x^2 + x - 6$: We need to find two numbers that multiply to $-6$ (the constant term) and add to $1$ (the coefficient of $x$). These numbers are $3$ and $-2$.

   Therefore, we can write: $x^2 + x - 6 = (x - 2)(x + 3)$

4. Combine the factored forms: Incorporating the GCF we factored out earlier, we have: $2(x - 2)(x + 3)$

4. Verify and Summarize

Let's verify our factorization by expanding $2(x - 2)(x + 3)$: $2[(x)(x) + (3x) - (2x) - (6)] = 2(x^2 + x - 6) = 2x^2 + 2x - 12$ The original expression is correctly factored.

Final Answer

The complete factorization of $2x^2 + 2x - 12$ is: $\boxed{2(x - 2)(x + 3)}$