We are asked to simplify the following two expressions: (e) $\frac{2x^2 + 17x + 21}{(x+2)(x^2-9)}$ (i) $\frac{3x - 10x - 24}{x(x^2 - 4)}$

1. Problem Description

We are asked to simplify the following two expressions:

(e) $\frac{2x^2 + 17x + 21}{(x+2)(x^2-9)}$

(i) $\frac{3x - 10x - 24}{x(x^2 - 4)}$

2. Solution Steps

(e)

First, we factor the numerator: $2x^2 + 17x + 21 = (2x + 3)(x + 7)$.

Next, we factor the denominator: $(x+2)(x^2-9) = (x+2)(x-3)(x+3)$.

Thus, we have:

$\frac{2x^2 + 17x + 21}{(x+2)(x^2-9)} = \frac{(2x+3)(x+7)}{(x+2)(x-3)(x+3)}$.

The expression is already simplified.

(i)

First, simplify the numerator: $3x - 10x - 24 = -7x - 24$.

Next, factor the denominator: $x(x^2 - 4) = x(x-2)(x+2)$.

Thus, we have:

$\frac{3x - 10x - 24}{x(x^2 - 4)} = \frac{-7x - 24}{x(x-2)(x+2)}$.

The expression is already simplified.

3. Final Answer

(e) $\frac{(2x+3)(x+7)}{(x+2)(x-3)(x+3)}$

(i) $\frac{-7x - 24}{x(x-2)(x+2)}$