To solve [tex]\(\frac{x^2 + 7x + 12}{x^3 - 9} \div \frac{x^2 + 10x + 24}{x^2 - 3x - 54}\)[/tex], we need to perform the following steps:
1. Factor the numerators and denominators of both fractions.
- For the first numerator [tex]\(x^2 + 7x + 12\)[/tex]:
[tex]\[
x^2 + 7x + 12 = (x + 3)(x + 4)
\][/tex]
- For the first denominator [tex]\(x^3 - 9\)[/tex]:
[tex]\[
x^3 - 9 = (x - 3)(x^2 + 3x + 9)
\][/tex]
- For the second numerator [tex]\(x^2 + 10x + 24\)[/tex]:
[tex]\[
x^2 + 10x + 24 = (x + 4)(x + 6)
\][/tex]
- For the second denominator [tex]\(x^2 - 3x - 54\)[/tex]:
[tex]\[
x^2 - 3x - 54 = (x - 9)(x + 6)
\][/tex]
2. Rewrite the original expression using the factored forms:
[tex]\[
\frac{(x + 3)(x + 4)}{(x - 3)(x^2 + 3x + 9)} \div \frac{(x + 4)(x + 6)}{(x - 9)(x + 6)}
\][/tex]
3. Convert the division to multiplication by taking the reciprocal of the second fraction:
[tex]\[
\frac{(x + 3)(x + 4)}{(x - 3)(x^2 + 3x + 9)} \times \frac{(x - 9)(x + 6)}{(x + 4)(x + 6)}
\][/tex]
4. Simplify the expression by canceling out common factors:
[tex]\[
\frac{(x + 3) \cancel{(x + 4)} (x - 9) \cancel{(x + 6)}}{(x - 3) (x^2 + 3x + 9) \cancel{(x + 4)} \cancel{(x + 6)}}
\][/tex]
This simplifies to:
[tex]\[
\frac{(x + 3)(x - 9)}{(x - 3)(x^2 + 3x + 9)}
\][/tex]
5. State the restrictions on the variable: The restrictions come from the values that make any of the denominators zero. From the factored forms, we get the restrictions:
- From [tex]\(x-3\)[/tex], [tex]\(x+4\)[/tex], and [tex]\(x+6\)[/tex], which appear in the denominators and numerators that can cancel out, [tex]\(x \neq -4\)[/tex], [tex]\(x \neq -3\)[/tex], and [tex]\(x \neq -6\)[/tex].
- Additionally, from the original expression `x^2 + 3x + 9`, the roots can be found by solving [tex]\(x^2 + 3x + 9 = 0\)[/tex]. This quadratic equation does not factor nicely and solving it results in complex roots which do not impact the real number restrictions. Thus, from the simplification process and denominators, the restrictions are: [tex]\(x \neq -4\)[/tex], [tex]\(x \neq -3\)[/tex], [tex]\(x \neq 3\)[/tex], and [tex]\(x \neq -6\)[/tex].
Hence, the completely simplified expression and restrictions are:
[tex]\[
\frac{(x + 3)(x - 9)}{(x - 3)(x^2 + 3x + 9)}, \quad x \neq -4, x \neq -3, x \neq 3, x \neq -6
\][/tex]
Looking at the choices provided:
- The correct choice is [tex]\(\frac{x-9}{x+3}, x \neq-4, x \neq-3, x \neq 3, x \neq-6\)[/tex]
