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Multiply the rational expressions.

[tex]\[ \frac{10x - 15x^2}{9x^2 + 24x + 16} \cdot \frac{9x^2 + 18x + 8}{9x^2 - 4} \][/tex]

Numerator: [tex]$\square$[/tex]
Denominator: [tex]$\square$[/tex]

Enter your answer for the numerator and denominator as an expression or as a whole number. Do not use spaces.

Examples: [tex]$2x + 1, 4x(5x + 1), 3x^{\wedge}2, 3$[/tex], or [tex]$-4$[/tex].



Answer :

GinnyAnswer
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.

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