Question

In the following exercise, perform the indicated operation:

$\frac{3}{2x^2+5x+2}-\frac{1}{2x^2+3x+1}$

Short answer

$\frac{1}{\left(x+2\right)\left(x+1\right)}$

Step-by-step solution

Step 1. Given information.

The following expression is given:

$\frac{3}{2x^2+5x+2}-\frac{1}{2x^2+3x+1}$

Step 2. Performing the subtraction operation.

Factor the denominator:

$\frac{3}{\left(2x+1\right)\left(x+2\right)}-\frac{1}{\left(2x+1\right)\left(x+1\right)}$

Find out the LCD:

$\left(2x+1\right)\left(x+2\right)\left(x+1\right)$

Rewrite the expression with LCD at the denominator:

$\frac{3\left(x+1\right)}{\left(2x+1\right)\left(x+2\right)\left(x+1\right)}-\frac{1\left(x+2\right)}{\left(2x+1\right)\left(x+1\right)\left(x+2\right)}$

Simplify:

$$\begin{gathered} \frac{3\left(x+1\right)-\left(x+2\right)}{\left(2x+1\right)\left(x+2\right)\left(x+1\right)} \\ \frac{3x+3-x-2}{\left(2x+1\right)\left(x+2\right)\left(x+1\right)} \\ \frac{2x+1}{\left(2x+1\right)\left(x+2\right)\left(x+1\right)} \end{gathered}$$

Cancel the common factors from the numerator and the denominator:

$\frac{1}{\left(x+2\right)\left(x+1\right)}$