We are asked to factor the quadratic $2x^2 + 12x + 16$ in two different ways, without starting from scratch for the second factoring.

1. Problem Description

We are asked to factor the quadratic $2x^2 + 12x + 16$ in two different ways, without starting from scratch for the second factoring.

2. Solution Steps

First, we factor the quadratic $2x^2 + 12x + 16$ by taking out the common factor of 2:

$2x^2 + 12x + 16 = 2(x^2 + 6x + 8)$

Now, we factor the quadratic $x^2 + 6x + 8$. We are looking for two numbers that multiply to 8 and add to

6. Those numbers are 2 and

4. So, $x^2 + 6x + 8 = (x+2)(x+4)$.

Thus, $2x^2 + 12x + 16 = 2(x+2)(x+4)$. We can multiply the 2 into either factor to get a different factored form.

First way: Multiply the 2 into $(x+2)$ to get $2(x+2)(x+4) = (2x+4)(x+4)$.

Second way: Multiply the 2 into $(x+4)$ to get $2(x+2)(x+4) = (x+2)(2x+8)$.

3. Final Answer

$(2x+4)(x+4)$

$(x+2)(2x+8)$