## Simplifying the Expression: (x^2-8x+12)(x^2-16)/(x^2-36)(x^2-4)

This expression involves a product of four quadratic expressions, and it can be simplified by factoring and canceling common terms. Let's break down the steps:

### 1. Factoring the Quadratic Expressions

**(x^2-8x+12)**can be factored as**(x-6)(x-2)****(x^2-16)**is a difference of squares, factoring as**(x+4)(x-4)****(x^2-36)**is also a difference of squares, factoring as**(x+6)(x-6)****(x^2-4)**is a difference of squares, factoring as**(x+2)(x-2)**

### 2. Substituting the Factored Expressions

Now, substitute these factored expressions back into the original expression:

[(x-6)(x-2)(x+4)(x-4)] / [(x+6)(x-6)(x+2)(x-2)]

### 3. Canceling Common Terms

Observe that several terms appear in both the numerator and denominator. We can cancel these common terms:

( **(x-6)**(x-2)(x+4)(x-4) ) / ( **(x+6)**( **(x-6)** )(x+2)( **(x-2)** )

This leaves us with:

**(x+4)(x-4) / (x+6)(x+2)**

### 4. Final Simplified Expression

The simplified expression after canceling common terms is **(x+4)(x-4) / (x+6)(x+2)**. We can leave it in this factored form, or multiply it out to get **(x^2-16) / (x^2 + 8x + 12)**.

**Important Note:** This simplified expression is equivalent to the original one **except** for the values of x that make the original expression undefined, which are:

**x = 6****x = -6****x = 2****x = -2**

These values make the original expression undefined because they would lead to division by zero in the denominator.