To simplify the given expression, we need to perform the operations step by step. Let's start from the given expression:
[tex]\[
\frac{\frac{3x^2 - 4x - 15}{4x^2 - 15x - 4}}{\frac{9x^2 + 21x + 10}{4x^2 + 13x + 3}}
\][/tex]
First, we need to convert the division of fractions into multiplication by taking the reciprocal of the second fraction:
[tex]\[
\frac{3x^2 - 4x - 15}{4x^2 - 15x - 4} \times \frac{4x^2 + 13x + 3}{9x^2 + 21x + 10}
\][/tex]
Next, let's factorize each polynomial.
Factorizing the numerators and denominators:
1. Factorize \(3x^2 - 4x - 15\):
[tex]\[
3x^2 - 4x - 15 = (3x + 5)(x - 3)
\][/tex]
2. Factorize \(4x^2 - 15x - 4\):
[tex]\[
4x^2 - 15x - 4 = (4x + 1)(x - 4)
\][/tex]
3. Factorize \(9x^2 + 21x + 10\):
[tex]\[
9x^2 + 21x + 10 = (3x + 2)(3x + 5)
\][/tex]
4. Factorize \(4x^2 + 13x + 3\):
[tex]\[
4x^2 + 13x + 3 = (4x + 1)(x + 3)
\][/tex]
Now, substitute the factored forms back into the expression:
[tex]\[
\frac{(3x + 5)(x - 3)}{(4x + 1)(x - 4)} \times \frac{(4x + 1)(x + 3)}{(3x + 2)(3x + 5)}
\][/tex]
The expression can now be simplified by canceling out common factors from the numerator and the denominator. Specifically, \((4x + 1)\) and \((3x + 5)\) appear in both the numerator and the denominator:
[tex]\[
\frac{(3x + 5)(x - 3)}{(4x + 1)(x - 4)} \times \frac{(4x + 1)(x + 3)}{(3x + 2)(3x + 5)} = \frac{(x - 3)(x + 3)}{(x - 4)(3x + 2)}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{(x - 3)(x + 3)}{(x - 4)(3x + 2)}}
\][/tex]
