## Factoring and Simplifying the Expression: (x^2+8x+12)(x^3+5x^2-6x)

This expression involves multiplying two polynomials. To simplify it, we can factor both polynomials and then multiply the resulting expressions.

### Factoring the Polynomials:

**1. Factoring (x^2 + 8x + 12)**

- Find two numbers that add up to 8 and multiply to 12. These numbers are 6 and 2.
- Therefore, we can factor this expression as:
**(x + 6)(x + 2)**

**2. Factoring (x^3 + 5x^2 - 6x)**

- First, factor out the greatest common factor (GCF) which is x:
**x(x^2 + 5x - 6)** - Now, factor the quadratic expression inside the parentheses. Find two numbers that add up to 5 and multiply to -6. These numbers are 6 and -1.
- Therefore, we can factor this expression as:
**x(x + 6)(x - 1)**

### Multiplying the Factored Expressions:

Now that we have factored both polynomials, we can multiply them together:

**(x + 6)(x + 2) * x(x + 6)(x - 1)**

We can rearrange the factors for easier multiplication:

**x(x + 6)(x + 6)(x + 2)(x - 1)**

This simplifies to:

**x(x + 6)^2 (x + 2)(x - 1)**

### Final Simplified Expression:

Therefore, the simplified expression for (x^2 + 8x + 12)(x^3 + 5x^2 - 6x) is: **x(x + 6)^2 (x + 2)(x - 1)**