Question

form. $\frac{3x^{2}+12x}{3x^{2}-75}cdot\frac{x^{2}+x - 20}{3x^{2}-12x}$
$\frac{x(x + 4)}{(x - 5)}cdot\frac{(x + 5)}{3}=\frac{x(x + 4)}{3(x - 5)}$
$\frac{(x + 4)}{3(x + 5)}cdot\frac{(x + 4)}{(x - 4)}=\frac{(x + 4)(x - 4)}{3(x + 5)(x - 4)}$
$\frac{3x}{(x - 5)}cdot\frac{(x + 5)}{1}=\frac{3x(x + 5)}{x - 5}$
$\frac{(x + 4)}{3(x - 5)}cdot\frac{1}{1}=\frac{(x + 4)}{3(x - 5)}$
steps: 1. factor all terms 2. reduce common terms 3. multiply & write in simplest factored form

Explanation:

Step1: Factor numerators and denominators

• For $\frac{3x^{2}+12x}{3x^{2}-75}$, factor out common factors:
• $3x^{2}+12x = 3x(x + 4)$
• $3x^{2}-75=3(x^{2}-25)=3(x + 5)(x - 5)$
• For $\frac{x^{2}+x - 20}{3x^{2}-12x}$, factor out common factors:
• $x^{2}+x - 20=(x + 5)(x - 4)$
• $3x^{2}-12x=3x(x - 4)$

So the original expression $\frac{3x^{2}+12x}{3x^{2}-75}\cdot\frac{x^{2}+x - 20}{3x^{2}-12x}=\frac{3x(x + 4)}{3(x + 5)(x - 5)}\cdot\frac{(x + 5)(x - 4)}{3x(x - 4)}$

Step2: Reduce common terms

Cancel out the common terms $3x$, $(x + 5)$ and $(x - 4)$ in the numerator and denominator.
We get $\frac{(x + 4)}{3(x - 5)}$

Answer:

$\frac{x + 4}{3(x - 5)}$