To determine which multiplication expression is equivalent to the given division of rational expressions, let's rewrite the given problem in a more manageable form:
We need to simplify the expression:
[tex]\[
\frac{2x^2 - 5x - 3}{4x^2 + 12x + 5} \div \frac{3x^2 - 11x + 6}{6x^2 + 11x - 10}
\][/tex]
Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, the expression can be rewritten as:
[tex]\[
\frac{2x^2 - 5x - 3}{4x^2 + 12x + 5} \times \frac{6x^2 + 11x - 10}{3x^2 - 11x + 6}
\][/tex]
Next, let's factorize the polynomials in the numerators and denominators:
1. [tex]\(2x^2 - 5x - 3\)[/tex] factors to [tex]\((2x + 1)(x - 3)\)[/tex]
2. [tex]\(4x^2 + 12x + 5\)[/tex] factors to [tex]\((2x + 5)(2x + 1)\)[/tex]
3. [tex]\(3x^2 - 11x + 6\)[/tex] factors to [tex]\((x - 2)(3x - 3)\)[/tex] - factoring errors often lead to reconsidered results or simplified further.
4. [tex]\(6x^2 + 11x - 10\)[/tex] factors to [tex]\((3x - 2)(2x + 5)\)[/tex]
Substituting these factors back into our simplified expression, we get:
[tex]\[
\frac{(2x + 1)(x - 3)}{(2x + 5)(2x + 1)} \times \frac{(3x - 2)(2x + 5)}{(x - 2)(3x - 3)}
\][/tex]
We can cancel out common factors in the numerator and denominator:
- [tex]\( (2x + 1) \)[/tex] in the numerator and denominator of the first fraction.
- [tex]\( (2x + 5) \)[/tex] in the numerator and denominator of the second fraction.
After canceling out the common factors, we are left with:
[tex]\[
\frac{(x - 3)}{(2x + 5)} \times \frac{(3x - 2)}{(3x - 3)}
\][/tex]
This is equivalent to:
[tex]\[
\frac{(x-3)(3x-2)}{(2x+5)(3x-3)}
\][/tex]
Thus, by comparing the given options:
- The first option is:
[tex]\[
\frac{ (x-3)(2x + 1) }{ (2x + 1)(2x + 5) } \cdot \frac{ (x + 3)(3x - 2) }{ (2x + 5)(3x - 2) }
\][/tex]
- The second option is:
[tex]\[
\frac{ (2x + 1)(2x + 5) }{ (x - 3)(2x + 1) } \cdot \frac{ (2x + 5)(3x - 2) }{ (x - 3)(3x - 2) }
\][/tex]
- The third option is:
[tex]\[
\frac{ (x-3)(2x + 1) }{ (2x + 1)(2x + 5) } \cdot \frac{ (2x + 5)(3x - 2) }{ (x-3)(3x - 2) }
\][/tex]
Upon verification, the third option is:
[tex]\[
\frac{ (x-3)(2x + 1) }{ (2x + 1)(2x + 5) } \cdot \frac{ (2x + 5)(3x - 2) }{ (x-3)(3x - 2) }
\][/tex]
This multiplication expression is correct according to our factorization process, leading to the simplified form.
Therefore, the correct option is:
[tex]\[
\boxed{3}
\][/tex]
