To solve the problem [tex]\(\frac{4 x^2-y^2}{8 x^2 + 10 x y + 3 y^2} \cdot \frac{4 x^2-9 x y-9 y^2}{2 x^2-5 x y-3 y^2}\)[/tex], let's break it down into steps and perform the simplification step-by-step.
1. Identify the numerators and denominators of each fraction:
The first fraction has:
- Numerator: [tex]\(4 x^2 - y^2\)[/tex]
- Denominator: [tex]\(8 x^2 + 10 x y + 3 y^2\)[/tex]
The second fraction has:
- Numerator: [tex]\(4 x^2 - 9 x y - 9 y^2\)[/tex]
- Denominator: [tex]\(2 x^2 - 5 x y - 3 y^2\)[/tex]
2. Multiply the numerators together:
[tex]\[
(4 x^2 - y^2) \cdot (4 x^2 - 9 x y - 9 y^2)
\][/tex]
3. Multiply the denominators together:
[tex]\[
(8 x^2 + 10 x y + 3 y^2) \cdot (2 x^2 - 5 x y - 3 y^2)
\][/tex]
4. Form the new fraction:
[tex]\[
\frac{(4 x^2 - y^2) \cdot (4 x^2 - 9 x y - 9 y^2)}{(8 x^2 + 10 x y + 3 y^2) \cdot (2 x^2 - 5 x y - 3 y^2)}
\][/tex]
5. Simplify the resulting fraction:
After performing the multiplications and simplifying the fraction, we obtain:
[tex]\[
\frac{(4 x^2 - y^2) \cdot (4 x^2 - 9 x y - 9 y^2)}{(8 x^2 + 10 x y + 3 y^2) \cdot (2 x^2 - 5 x y - 3 y^2)} = \frac{2x - y}{2x + y}
\][/tex]
Therefore, the final simplified result of the operation is:
[tex]\[
\boxed{\frac{2x - y}{2x + y}}
\][/tex]
