Question

(\frac{5x^{2} + 13x + 6}{-2x - 12} cdot \frac{x^{2} + 8x + 12}{25x^{2} + 30x + 9})

Explanation:

Step1: Factor all polynomials

1. $5x^2+13x+6=(5x+3)(x+2)$
2. $-2x-12=-2(x+6)$
3. $x^2+8x+12=(x+2)(x+6)$
4. $25x^2+30x+9=(5x+3)^2$

Step2: Substitute factored forms

$$\frac{(5x+3)(x+2)}{-2(x+6)} \cdot \frac{(x+2)(x+6)}{(5x+3)^2}$$

Step3: Cancel common factors

Cancel $(5x+3)$, $(x+6)$ from numerator/denominator:
$$\frac{(x+2)}{-2} \cdot \frac{(x+2)}{(5x+3)}$$

Step4: Multiply remaining terms

$$\frac{(x+2)^2}{-2(5x+3)} = -\frac{(x+2)^2}{2(5x+3)}$$

Answer:

$-\frac{(x+2)^2}{2(5x+3)}$ (or equivalently $\frac{(x+2)^2}{-10x-6}$)