Certainly! Let's start by analyzing the given expression and simplifying it step-by-step to determine which of the provided options is equivalent to it.
The given expression is:
[tex]\[ \frac{x+3}{x+4} \cdot \frac{1}{x^2-9} \][/tex]
First, let's rewrite the expression with a single fraction by multiplying the numerators and the denominators:
[tex]\[ \frac{(x+3) \cdot 1}{(x+4) \cdot (x^2-9)} = \frac{x+3}{(x+4)(x^2-9)} \][/tex]
Next, we observe that [tex]\(x^2 - 9\)[/tex] can be factored:
[tex]\[ x^2 - 9 = (x-3)(x+3) \][/tex]
So, we can substitute this factorization into our expression:
[tex]\[ \frac{x+3}{(x+4) \cdot (x-3)(x+3)} \][/tex]
Notice that [tex]\((x+3)\)[/tex] in the numerator and denominator can cancel each other out, because [tex]\(x \neq -3\)[/tex] (to avoid division by zero). After canceling, we are left with:
[tex]\[ \frac{1}{(x+4)(x-3)} \][/tex]
Now let's compare this with the given options:
A. [tex]\( \frac{1}{x^2+x-12} \)[/tex]
- [tex]\(x^2 + x - 12 \)[/tex] does not factor to [tex]\((x+4)(x-3)\)[/tex].
B. [tex]\( \frac{1}{x^2+7x+12} \)[/tex]
- [tex]\(x^2 + 7x + 12 \)[/tex] does not factor to [tex]\((x+4)(x-3)\)[/tex].
C. [tex]\( \frac{x+3}{x^2 + x - 12} \)[/tex]
- This option does not simplify to [tex]\( \frac{1}{(x+4)(x-3)}\)[/tex].
D. [tex]\( \frac{x+3}{x^2 + 7x + 12} \)[/tex]
- [tex]\( x^2 + 7x + 12 = (x+3)(x+4) \)[/tex]. However, [tex]\(x^2 + 7x + 12\)[/tex] does not match our factored denominator [tex]\((x+4)(x-3)\)[/tex].
Upon examining all options, none of them exactly matches our result directly except for understanding that the denominator does not align with option D precisely after canceling the factors.
However, considering the results and the structure of the answer:
The correct expression reduced to favourable [tex]\( \frac{1}{(x+4)(x-3)} \)[/tex] matches our detailed simplified form. Therefore, in simplified and match parallels between our results:
The expressions do not exactly match any among our given multiple choices without further rearrangement (none among the four coincide fully simple factor comparisons).
However, among reductions and cancellations closely aligns:
The simplified form considered closer [tex]\(\boxed{\frac{1}{(x-3)(x+4)}}\)[/tex] favors to the closest yet not displayed among options accurately. however in taking strict deficiency among tally - none coincide as exact beyond additional rearrangements.
