Question

(\frac{2x^{4}-6x^{3}}{x^{2}-9}cdot\frac{x^{2}+5x + 6}{10x^{5}}=)

Explanation:

Step1: Factor all polynomials

• Numerator 1: $2x^4 - 6x^3 = 2x^3(x - 3)$
• Denominator 1: $x^2 - 9 = (x - 3)(x + 3)$
• Numerator 2: $x^2 + 5x + 6 = (x + 2)(x + 3)$
• Denominator 2: $10x^5 = 2 \cdot 5 \cdot x^5$

Rewrite the expression:
$$\frac{2x^3(x - 3)}{(x - 3)(x + 3)} \cdot \frac{(x + 2)(x + 3)}{10x^5}$$

Step2: Cancel common factors

Cancel $(x-3)$, $(x+3)$, $2$, and $x^3$ from numerator and denominator:
$$\frac{1}{1} \cdot \frac{(x + 2)}{5x^2}$$

Step3: Multiply remaining terms

Multiply the simplified fractions:
$$\frac{x + 2}{5x^2}$$

Answer:

$\boldsymbol{\frac{x + 2}{5x^2}}$ (where $x
eq -3, 0, 3$)