Question

divide the rational expressions. write your answer in simplest form. $\frac{x + 3}{x^{2}+4x + 3}div\frac{10x^{2}}{20x^{2}+20x}$ $\frac{2}{x}$ $\frac{x}{2}$ $\frac{1}{2x(x + 1)^{2}}$ steps: 1. multiply by reciprocal 2. factor all terms 3. reduce common terms 3. write in simplest factored form

Explanation:

Step1: Multiply by reciprocal

$\frac{x + 3}{x^{2}+4x + 3}\div\frac{10x^{2}}{20x^{2}+20x}=\frac{x + 3}{x^{2}+4x + 3}\times\frac{20x^{2}+20x}{10x^{2}}$

Step2: Factor the expressions

Factor $x^{2}+4x + 3=(x + 1)(x+3)$, and $20x^{2}+20x = 20x(x + 1)$. So the expression becomes $\frac{x + 3}{(x + 1)(x + 3)}\times\frac{20x(x + 1)}{10x^{2}}$

Step3: Reduce common terms

Cancel out the common terms $(x + 3)$ and $(x + 1)$. We get $\frac{1}{1}\times\frac{20x}{10x^{2}}$

Step4: Simplify the result

$\frac{20x}{10x^{2}}=\frac{2}{x}$

Answer:

$\frac{2}{x}$