Question

In the following exercises, multiply the rational expressions.$\frac{5c^2+9c+2}{c^2-25}.\frac{c^2+10c+25}{3c^2-14c-5}$

Short answer

The simplified expression is$\frac{(5c^2+9c+2)\left(c+5\right)}{\left(3c+1\right){\left(c-5\right)}^2}.$

Step-by-step solution

Step 1. Given Information.

$\frac{5c^2+9c+2}{c^2-25}.\frac{c^2+10c+25}{3c^2-14c-5}$

Step 2. Factor each numerator and denominator completely 

$$\begin{gathered} c^2+10c+25=\left(c+5\right)\left(c+5\right) \\ {c}^2-25=\left(c-5\right)\left(c+5\right) \\ 3c^2-14c-5=\left(3c+1\right)\left(c-5\right) \end{gathered}$$

Then, multiply the numerators and denominators.

$$\begin{gathered} \frac{5c^2+9c+2}{c^2-25}.\frac{c^2+10c+25}{3c^2-14c-5} \\ =\frac{5c^2+9c+2}{\left(c-5\right)\left(c+5\right)}.\frac{\left(c+5\right)\left(c+5\right)}{\left(3c+1\right)\left(c-5\right)} \end{gathered}$$

Step 3. Simplify by dividing out common factors 

$\frac{(5c^2+9c+2)\left(c+5\right)}{\left(3c+1\right){\left(c-5\right)}^2}.$