Answer
The solutions are $m = -1$ and $m = -3$.
Work Step by Step
The first thing we want to do is get rid of the fractions. We do this by multiplying both sides by the denominator of the fraction we want to eliminate:
$3m(m + 4) = -9$
Use the distributive property:
$3m^2 + 12m = -9$
Collect all terms on the left side of the equation:
$3m^2 + 12m + 9 = 0$
Factor out what is common to all the terms:
$3(m^2 + 4m + 3) = 0$
We see that we have a quadratic expression inside the parentheses. Quadratic expressions 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 $3$, but when added together will give us the $b$ term, which is $4$. This means that both factors are positive.
Let's look at possible factors:
$3$ and $1$
Let's split the middle term:
$m^2 + 3m + m + 3$
Group the first two terms and the last two terms, paying attention to the signs:
$(m^2 + 3m) + (m + 3)$
Factor out what is common in both groups:
$m(m + 3) + (m + 3)$
Group the factors:
$(m + 1)(m + 3)$
We can now rewrite the equation incorporating these factors in place of the quadratic expression:
$3(m + 1)(m + 3) = 0$
Divide both sides by $3$:
$(m + 1)(m + 3) = 0$
According to the zero product property, if the product of two factors equals $0$, then either factor can be $0$; therefore, we can set each of these factors equal to $0$ and solve.
Let's look at the first factor:
$m + 1 = 0$
Subtract $1$ from each side of the equation:
$m = -1$
Let's look at the second factor:
$m + 3 = 0$
Subtract $3$ from each side:
$m = -3$
The solutions are $m = -1$ and $m = -3$.
To check our answers, we plug the values into the original equation.
Let's check $m = -1$:
$3(-1) = \frac{-9}{(-1) + 4}$
Multiply:
$-3 = \frac{-9}{3}$
Simplify the fraction:
$-3 = -3$
This solution is correct. Let's check the other solution:
$3(-3) = \frac{-9}{(-3) + 4}$
Multiply:
$-9 = \frac{-9}{1}$
Simplify the fraction:
$-9 = -9$
This solution is also correct.
