Certainly! Let's perform the indicated operation and simplify the result step-by-step.
We start with the given fractions:
[tex]\[
\frac{3 a^2 - 13 a + 4}{9 a^2 - 6 a + 1} \cdot \frac{28 + 7 a}{a^2 - 16}
\][/tex]
1. Factor the numerator and denominator of both fractions:
For the first fraction:
- Numerator: [tex]\(3 a^2 - 13 a + 4\)[/tex]
- Denominator: [tex]\(9 a^2 - 6 a + 1\)[/tex]
The factors are:
[tex]\[
3 a^2 - 13 a + 4 = (a - 4)(3 a - 1)
\][/tex]
and
[tex]\[
9 a^2 - 6 a + 1 = (3 a - 1)^2
\][/tex]
For the second fraction:
- Numerator: [tex]\(28 + 7 a\)[/tex]
- Denominator: [tex]\(a^2 - 16\)[/tex]
The factors are:
[tex]\[
28 + 7 a = 7(a + 4)
\][/tex]
and
[tex]\[
a^2 - 16 = (a - 4)(a + 4)
\][/tex]
2. Substitute the factored forms into the original expression:
[tex]\[
\frac{(a - 4)(3 a - 1)}{(3 a - 1)^2} \cdot \frac{7(a + 4)}{(a - 4)(a + 4)}
\][/tex]
3. Combine the fractions and simplify:
[tex]\[
\frac{(a - 4)(3 a - 1) \cdot 7(a + 4)}{(3 a - 1)^2 \cdot (a - 4)(a + 4)}
\][/tex]
Notice that some terms in the numerator and denominator can be cancelled out:
- [tex]\((a - 4)\)[/tex] in both the numerator and the denominator
- [tex]\((a + 4)\)[/tex] in both the numerator and the denominator
After cancelling these terms, we get:
[tex]\[
\frac{7 (3 a - 1)}{(3 a - 1)^2}
\][/tex]
4. Simplify the remaining fraction by cancelling out [tex]\((3 a - 1)\)[/tex]:
[tex]\[
\frac{7}{3 a - 1}
\][/tex]
Thus, the final simplified result of the operation is:
[tex]\[
\boxed{\frac{7}{3 a - 1}}
\][/tex]
