(x%5E2%2Bx-8)%2B12.jpeg "(x^2+x)(x^2+x-8)+12")
Factoring the Expression (x^2+x)(x^2+x-8)+12
This article will guide you through the process of factoring the expression (x^2+x)(x^2+x-8)+12.
1. Recognizing a Pattern
Notice that the expression has a repeating pattern: (x^2+x). Let's simplify things by using substitution.
- Let y = x^2 + x.
Now our expression becomes: y(y-8) + 12
2. Expanding and Simplifying
Expand the expression:
- y^2 - 8y + 12
3. Factoring the Quadratic
The expression is now a simple quadratic equation. We need to find two numbers that add up to -8 and multiply to 12. These numbers are -6 and -2.
- (y - 6)(y - 2)
4. Substituting Back
Now substitute back y = x^2 + x:
- (x^2 + x - 6)(x^2 + x - 2)
5. Factoring Further
The expressions in the parentheses are also quadratic expressions that can be factored:
- (x+3)(x-2)(x+2)(x-1)
Final Factored Expression
Therefore, the completely factored form of the expression (x^2+x)(x^2+x-8)+12 is:
(x+3)(x-2)(x+2)(x-1)