Sure, let's go through the solution step-by-step.
1. Factor the Numerator and Denominator of the First Expression:
[tex]\[
\frac{a^2 + 8a + 15}{a^2 + 7a + 12}
\][/tex]
The numerator [tex]\(a^2 + 8a + 15\)[/tex] can be factored as:
[tex]\[
(a + 3)(a + 5)
\][/tex]
The denominator [tex]\(a^2 + 7a + 12\)[/tex] can be factored as:
[tex]\[
(a + 3)(a + 4)
\][/tex]
So, the first expression becomes:
[tex]\[
\frac{(a + 3)(a + 5)}{(a + 3)(a + 4)}
\][/tex]
2. Factor the Numerator and Denominator of the Second Expression:
[tex]\[
\frac{3a + 12}{a^2 - 25}
\][/tex]
The numerator [tex]\(3a + 12\)[/tex] can be factored as:
[tex]\[
3(a + 4)
\][/tex]
The denominator [tex]\(a^2 - 25\)[/tex] can be factored as:
[tex]\[
(a - 5)(a + 5)
\][/tex]
So, the second expression becomes:
[tex]\[
\frac{3(a + 4)}{(a - 5)(a + 5)}
\][/tex]
3. Multiply the Two Factored Expressions:
[tex]\[
\left(\frac{(a + 3)(a + 5)}{(a + 3)(a + 4)}\right) \cdot \left(\frac{3(a + 4)}{(a - 5)(a + 5)}\right)
\][/tex]
4. Cancel Out Common Factors:
- The [tex]\((a + 3)\)[/tex] terms cancel each other out.
- The [tex]\((a + 4)\)[/tex] terms cancel each other out.
- The [tex]\((a + 5)\)[/tex] terms cancel each other out.
We are left with:
[tex]\[
\frac{3}{(a - 5)}
\][/tex]
5. Simplified Result:
[tex]\[
\frac{3}{a - 5}
\][/tex]
Therefore, the simplified product of the given expressions is:
[tex]\[
\frac{3}{a - 5}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\frac{3}{a - 5}}
\][/tex]
