To multiply the given rational expressions, we'll follow these steps:
Given rational expressions:
[tex]\[
\frac{10x - 15x^2}{9x^2 + 24x + 16} \cdot \frac{9x^2 + 18x + 8}{9x^2 - 4}
\][/tex]
1. Multiply the numerators together:
[tex]\[
(10x - 15x^2) \cdot (9x^2 + 18x + 8)
\][/tex]
2. Multiply the denominators together:
[tex]\[
(9x^2 + 24x + 16) \cdot (9x^2 - 4)
\][/tex]
3. Expand and simplify (if possible) the resulting polynomials:
- Numerator:
[tex]\[
(10x - 15x^2) \cdot (9x^2 + 18x + 8) = -135x^4 - 180x^3 + 60x^2 + 80x
\][/tex]
- Denominator:
[tex]\[
(9x^2 + 24x + 16) \cdot (9x^2 - 4) = 81x^4 + 216x^3 + 108x^2 - 96x - 64
\][/tex]
The final expressions are:
[tex]\[
\text{Numerator: } -135x^4 - 180x^3 + 60x^2 + 80x
\][/tex]
[tex]\[
\text{Denominator: } 81x^4 + 216x^3 + 108x^2 - 96x - 64
\][/tex]
These are the final simplified forms of the numerator and the denominator after multiplying the given rational expressions.
