## Factoring and Expanding (x^2+7x-12)(x^2-9x+1)

This expression represents the product of two quadratic expressions. To simplify it, we can either expand it directly or factor each quadratic expression first and then multiply. Let's explore both methods:

**1. Expanding Directly**

To expand the expression, we can use the distributive property (or FOIL method):

**Multiply each term of the first quadratic by each term of the second quadratic:**- (x^2) * (x^2) + (x^2) * (-9x) + (x^2) * (1) + (7x) * (x^2) + (7x) * (-9x) + (7x) * (1) + (-12) * (x^2) + (-12) * (-9x) + (-12) * (1)

**Simplify by combining like terms:**- x^4 - 9x^3 + x^2 + 7x^3 - 63x^2 + 7x - 12x^2 + 108x - 12

**Combine the coefficients of each term:**- x^4 - 2x^3 - 74x^2 + 115x - 12

Therefore, the expanded form of (x^2+7x-12)(x^2-9x+1) is **x^4 - 2x^3 - 74x^2 + 115x - 12**.

**2. Factoring First**

We can factor each quadratic expression separately:

**Factoring x^2+7x-12:**- Find two numbers that multiply to -12 and add up to 7. These numbers are 12 and -1.
- Therefore, x^2+7x-12 =
**(x+12)(x-1)**.

**Factoring x^2-9x+1:**- This quadratic cannot be factored easily using integers. We can leave it as it is.

Now, we can multiply the factored expressions:

- (x^2+7x-12)(x^2-9x+1) =
**(x+12)(x-1)(x^2-9x+1)**

This form is not simplified but represents the factored form of the original expression.

**In Conclusion**

Both methods lead to the same result, just in different forms. The expanded form is useful for further manipulation or finding specific coefficients, while the factored form might be helpful for identifying roots or understanding the behavior of the expression.