Answer
$\frac{r - 3}{r + 7}$
Work Step by Step
To add or subtract rational expressions, we need to make sure that the expressions have the same denominator. They both have the same denominator.
We need to make sure that each denominator is factored completely.
We see that the denominators are quadratic expressions, which are given by the formula:
$ax^2 + bx + c$, where $a$, $b$, and $c$ are all real numbers.
To factor the expression in the denominator of the first fraction, we want to find which factors when multiplied will give us the product of the $a$ and $c$ terms, which is $63$, but when added together will give us the $b$ term, which is $16$. This means that both factors are positive.
Let's look at possible factors:
$9$ and $7$
$21$ and $3$
It looks like the first combination will work. Let's split the middle term:
$r^2 + 9r + 7r + 63$
Group the first two terms and the last two terms:
$(r^2 + 9r) + (7r + 63)$
Factor out what is common in both groups:
$r(r + 9) + 7(r + 9)$
Group the factors:
$(r + 7)(r + 9)$
The exercise can now be rewritten as:
$\frac{r^2 + 10r}{(r + 7)(r + 9)} - \frac{4r + 27}{(r + 7)(r + 9)}$
Rewrite the two fractions as one with the same denominator:
$\frac{(r^2 + 10r) - (4r + 27)}{(r + 7)(r + 9)}$
Combine like terms in the numerator:
$\frac{r^2 + 6r - 27)}{(r + 7)(r + 9)}$
Factor the quadratic expression in the numerator:
$\frac{(r + 9)(r - 3)}{(r + 7)(r + 9)}$
Cancel the $r + 9$ factor in the numerator and denominator:
$\frac{r - 3}{r + 7}$
