Answer
$\frac{a + 2}{(a + 3)(a - 3)(a - 1)}$
Work Step by Step
To multiply rational expressions, 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 $9$, but when added together will give us the $b$ term, which is $6$. This means that both factors are positive.
Let's look at possible factors:
$3$ and $3$
$9$ and $1$
It looks like the first combination will work. Let's write the expression in factor form:
$(a + 3)(a + 3)$
To factor the expression in the denominator of the second fraction, we want to find which factors when multiplied will give us the product of the $a$ and $c$ terms, which is $3$, but when added together will give us the $b$ term, which is $-4$. This means that both factors are negative.
Let's look at possible factors:
$-3$ and $-1$
Let's rewrite the expression in factor form:
$(a - 3)(a - 1)$
The exercise can now be rewritten as:
$\frac{(a + 2)(a + 3)}{(a + 3)(a + 3)(a - 3)(a - 1)}$
Cancel out the $a + 3$ terms in the numerator and denominator:
$\frac{a + 2}{(a + 3)(a - 3)(a - 1)}$
