To solve the given expression [tex]\(\frac{x^2 + 7x + 12}{x^2 - 9} \div \frac{x^2 + 10x + 24}{x^2 - 3x - 54}\)[/tex] and simplify it completely, we can follow these steps:
1. Understand the Division of Fractions:
When dividing fractions, we multiply by the reciprocal of the second fraction. Therefore, we can rewrite the problem as:
[tex]\[
\frac{x^2 + 7x + 12}{x^2 - 9} \times \frac{x^2 - 3x - 54}{x^2 + 10x + 24}
\][/tex]
2. Factorize the Numerators and Denominators:
Let's factorize each part of the expression:
[tex]\[
x^2 + 7x + 12 = (x + 3)(x + 4)
\][/tex]
[tex]\[
x^2 - 9 = (x + 3)(x - 3)
\][/tex]
[tex]\[
x^2 + 10x + 24 = (x + 4)(x + 6)
\][/tex]
[tex]\[
x^2 - 3x - 54 = (x - 9)(x + 6)
\][/tex]
3. Substitute the Factored Forms:
Replace each part of the original expression with its factored form:
[tex]\[
\frac{(x + 3)(x + 4)}{(x + 3)(x - 3)} \times \frac{(x - 9)(x + 6)}{(x + 4)(x + 6)}
\][/tex]
4. Simplify the Expression by Canceling Common Factors:
Observe and cancel out common factors in the numerators and denominators:
[tex]\[
\frac{(x + 3)(x + 4)}{(x + 3)(x - 3)} \times \frac{(x - 9)(x + 6)}{(x + 4)(x + 6)}
= \frac{\cancel{(x + 3)} \cancel{(x + 4)}}{\cancel{(x + 3)}(x - 3)} \times \frac{(x - 9)\cancel{(x + 6)}}{\cancel{(x + 4)} \cancel{(x + 6)}}
= \frac{x - 9}{x - 3}
\][/tex]
5. State Restrictions:
The restrictions on the variable [tex]\(x\)[/tex] come from the original denominators to avoid division by zero:
[tex]\[
x^2 - 9 = (x + 3)(x - 3) \quad \Rightarrow \quad x \neq \pm 3
\][/tex]
[tex]\[
x^2 + 10x + 24 = (x + 4)(x + 6) \quad \Rightarrow \quad x \neq -4, -6
\][/tex]
[tex]\[
x^2 - 3x - 54 = (x - 9)(x + 6) \quad \Rightarrow \quad x \neq 9, -6
\][/tex]
Combining these, the restrictions are:
[tex]\[
x \neq -4, -3, 3, -6
\][/tex]
The simplified expression and its restrictions are:
[tex]\[
\frac{x - 9}{x - 3}, \quad \text{where} \quad x \neq -4, -3, 3, -6
\][/tex]
So the correct choice is:
[tex]\[
\frac{x-9}{x-3}, \quad x \neq -4, x \neq -3, x \neq 3, x \neq -6
\][/tex]
