Question

multiply as indicated.

1. $\frac{5x}{3x+6} cdot \frac{x^2 - 2x - 8}{x^2 + 2x - 8}$
2. $\frac{x^2 - 5x + 6}{x - 3} cdot \frac{x + 3}{x^2 - 9}$

Explanation:

Step1: Factor all expressions

For problem 11:

• $3x+6 = 3(x+2)$
• $x^2-2x-8 = (x-4)(x+2)$
• $x^2+2x-8 = (x+4)(x-2)$

For problem 12:

• $x^2-5x+6 = (x-2)(x-3)$
• $x^2-9 = (x+3)(x-3)$

Step2: Write product & cancel terms

Problem 11:

$$\frac{5x}{3(x+2)} \cdot \frac{(x-4)(x+2)}{(x+4)(x-2)}$$
Cancel $(x+2)$:
$$\frac{5x(x-4)}{3(x+4)(x-2)}$$

Problem 12:

$$\frac{(x-2)(x-3)}{x-3} \cdot \frac{x+3}{(x+3)(x-3)}$$
Cancel $(x-3)$ and $(x+3)$:
$$\frac{x-2}{x-3}$$

Step3: Simplify remaining expressions

Problem 11:

Expand numerator: $5x(x-4)=5x^2-20x$
Denominator: $3(x+4)(x-2)=3(x^2+2x-8)=3x^2+6x-24$
(or leave in factored form)

Problem 12:

Expression is already fully simplified.

Answer:

1. $\boldsymbol{\frac{5x(x-4)}{3(x+4)(x-2)}}$ (or $\boldsymbol{\frac{5x^2-20x}{3x^2+6x-24}}$)
2. $\boldsymbol{\frac{x-2}{x-3}}$