(x^2+8x+12)(x^2+12x+32)+16

4 min read Jun 17, 2024

Factoring and Simplifying the Expression (x^2+8x+12)(x^2+12x+32)+16

This article will guide you through factoring and simplifying the expression (x^2+8x+12)(x^2+12x+32)+16. We'll break it down step by step to make the process clear.

Factoring the Quadratic Expressions

Let's start by factoring the quadratic expressions within the parentheses:

(x^2 + 8x + 12): This factors to (x + 2)(x + 6)
(x^2 + 12x + 32): This factors to (x + 4)(x + 8)

Now our expression looks like this: (x + 2)(x + 6)(x + 4)(x + 8) + 16

Rearranging for Easier Factoring

To make the expression easier to work with, let's rearrange it slightly:

(x + 2)(x + 8)(x + 4)(x + 6) + 16

Notice that we've grouped the terms with similar factors together.

Recognizing a Pattern

Now, observe the first two terms: (x + 2)(x + 8)(x + 4)(x + 6)

This looks very similar to a perfect square trinomial. Let's see if we can manipulate it to fit that pattern.

Consider the following:

(x + 2)(x + 8) is almost (x + 5)^2 (we just need an extra '3')
(x + 4)(x + 6) is almost (x + 5)^2 (we need an extra '1')

Let's add and subtract these extra values inside the expression:

(x + 2)(x + 8) + 3(x + 4)(x + 6) - 3(x + 4)(x + 6) + 16

We've added and subtracted 3(x + 4)(x + 6) which doesn't change the value of the expression.

Completing the Square

Now, we can group terms to complete the square:

[(x + 2)(x + 8) + 3(x + 4)(x + 6)] - 3(x + 4)(x + 6) + 16

[(x + 5)^2 - 1] + 3[(x + 5)^2 - 1] - 3(x + 4)(x + 6) + 16

4(x + 5)^2 - 4 - 3(x + 4)(x + 6) + 16

4(x + 5)^2 - 3(x + 4)(x + 6) + 12

Final Simplification

The expression is now partially factored. It is not possible to factor it further in a simple way.

Therefore, the simplified form of the expression is: 4(x + 5)^2 - 3(x + 4)(x + 6) + 12