# How do you add or subtract (x^2+2x)/(12x+54) - (3-x)/(8x+36)?

Apr 21, 2018

$\frac{x - 1}{12}$

#### Explanation:

Given: $\frac{{x}^{2} + 2 x}{12 x + 54} - \frac{3 - x}{8 x + 36}$

Factor both numerators and denominators:

$\frac{{x}^{2} + 2 x}{12 x + 54} - \frac{3 - x}{8 x + 36} = \frac{x \left(x + 2\right)}{6 \left(2 x + 9\right)} - \frac{3 - x}{4 \left(2 x + 9\right)}$

Find the common denominator and simplify:

$\frac{x \left(x + 2\right)}{6 \left(2 x + 9\right)} \cdot \left(\frac{2}{2}\right) - \frac{3 - x}{4 \left(2 x + 9\right)} \left(\cdot \frac{3}{3}\right)$

$\frac{2 x \left(x + 2\right)}{12 \left(2 x + 9\right)} - \frac{3 \left(3 - x\right)}{12 \left(2 x + 9\right)}$

$\frac{2 x \left(x + 2\right) - 3 \left(3 - x\right)}{12 \left(2 x + 9\right)}$

$\frac{2 {x}^{2} + 4 x - 9 + 3 x}{12 \left(2 x + 9\right)}$

$\frac{2 {x}^{2} + 7 x - 9}{12 \left(2 x + 9\right)}$

Factor the quadratic in the numerator:
$\frac{\left(2 x + 9\right) \left(x - 1\right)}{12 \left(2 x + 9\right)}$

Cancel any common factors:
$\frac{\cancel{\left(2 x + 9\right)} \left(x - 1\right)}{12 \cancel{\left(2 x + 9\right)}}$

$= \frac{x - 1}{12}$