Answer
$\frac{-2(7m + 37)}{(m + 7)(m + 1)}$
Work Step by Step
To add or subtract rational expressions, we need to make sure that the expressions have the same denominator.
We need to rewrite the two expressions with the same denominator, so we need to find the least common denominator (LCD) for both expressions.
The first thing we want to do is to make sure that each quadratic expression is factored completely. The second expression is, but the first is not.
We see that the numerator and denominator in the first expression are quadratic; therefore, they follow 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 $7$, but when added together will give us the $b$ term, which is $8$. This means that both factors should be positive.
Let's look at possible factors:
$7$ and $1$
Let's rewrite the expression in factor form:
$(m + 7)(m + 1)$
The expression in the numerator is a special type of quadratic equation; it is called the difference of squares and follows the formula $ax^2 - b^2$, where $a$ and $b$ are real numbers. The factored form is $(a - b)(a + b)$.
In this case, $a = m$ and $b = 5$.
The factored expression is:
$(m - 5)(m + 5)$
The exercise can now be rewritten as:
$\frac{(m - 5)(m + 5)}{(m + 7)(m + 1)} - \frac{m + 7}{m + 1}$
Next, we want to find the least common denominator (LCD). We do this by taking the highest power of each factor in the denominators of the fractions. Before we can do that, we need to factor the denominators:
LCD = $(m + 7)(m + 1)$
Now that we have the least common denominator, we multiply the numerator of each fraction with the factor or factors it is missing in its denominator:
$\frac{(m - 5)(m + 5)}{(m + 7)(m + 1)} - \frac{(m + 7)(m + 7)}{(m + 7)(m + 1)}$
Rewrite the two fractions as one with the same denominator:
$\frac{(m - 5)(m + 5) - (m + 7)(m + 7)}{(m + 7)(m + 1)}$
Use the distributive property to rewrite the numerator:
$\frac{(m^2 - 25) - (m^2 + 14m + 49)}{(m + 7)(m + 1)}$
Combine like terms in the numerator:
$\frac{-14m - 74}{(m + 7)(m + 1)}$
Factor out what is common in the numerator:
$\frac{-2(7m + 37)}{(m + 7)(m + 1)}$
