# Question #10b08

Apr 4, 2017

Your answer is incorrect. A correct simplification could include $- \frac{3 x + 2}{{x}^{2} - 4}$.

#### Explanation:

As with most fractions, we should try to find a common denominator.

The first fraction's denominator can be factored as a difference of squares:

${x}^{2} / \left({x}^{2} - 4\right) - \frac{x + 1}{x - 2} = {x}^{2} / \left(\left(x + 2\right) \left(x - 2\right)\right) - \frac{x + 1}{x - 2}$

We now should see that the least common denominator is $\left(x + 2\right) \left(x - 2\right)$. So, we need to multiply the second fraction by $\left(x + 2\right)$ in the numerator and denominator:

${x}^{2} / \left(\left(x + 2\right) \left(x - 2\right)\right) - \frac{x + 1}{x - 2} = {x}^{2} / \left(\left(x + 2\right) \left(x - 2\right)\right) - \frac{\left(x + 1\right) \left(x + 2\right)}{\left(x + 2\right) \left(x - 2\right)}$

Simplify the numerator of the second fraction by FOILing $\left(x + 1\right) \left(x + 2\right)$:

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

Since the fractions have the same denominator, we can combine the numerators. Be careful, though--since the second fraction is being subtracted, we will have to use parentheses and subtract the entire numerator of the second denominator:

${x}^{2} / \left(\left(x + 2\right) \left(x - 2\right)\right) - \frac{{x}^{2} + 3 x + 2}{\left(x + 2\right) \left(x - 2\right)} = \frac{{x}^{2} - \left({x}^{2} + 3 x + 2\right)}{\left(x + 2\right) \left(x - 2\right)}$

Distributing the negative:

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

Canceling:

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

This could also be written as any of the following:

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