Let's multiply the given rational expressions and simplify the result step by step.
The given expressions are:
[tex]\[
\frac{6 m^2}{m^2 + 13 m + 36} \cdot \frac{m^2 - 16}{18 m}
\][/tex]
### Step 1: Factor the polynomials
First, we need to factor the quadratic expressions in the numerators and denominators wherever possible.
1. The denominator [tex]\( m^2 + 13 m + 36 \)[/tex]:
[tex]\[
m^2 + 13 m + 36 = (m + 4)(m + 9)
\][/tex]
2. The numerator [tex]\( m^2 - 16 \)[/tex]:
[tex]\[
m^2 - 16 = (m - 4)(m + 4)
\][/tex]
(This is the difference of squares.)
### Step 2: Rewrite the expressions with the factored forms
Now we rewrite the original problem using these factored forms:
[tex]\[
\frac{6 m^2}{(m + 4)(m + 9)} \cdot \frac{(m - 4)(m + 4)}{18 m}
\][/tex]
### Step 3: Multiply the fractions
To multiply the fractions, we multiply the numerators together and the denominators together:
[tex]\[
\frac{6 m^2 (m - 4)(m + 4)}{(m + 4)(m + 9) \cdot 18 m}
\][/tex]
### Step 4: Simplify the expression
Now, we look for common factors in the numerator and denominator that can be cancelled out to simplify the expression:
- The factor [tex]\(m + 4\)[/tex] appears in both the numerator and the denominator.
- The factor [tex]\(m\)[/tex] appears in both the numerator ([tex]\(6 m^2\)[/tex]) and the denominator ([tex]\(18 m\)[/tex]).
After cancelling the common factors:
[tex]\[
\frac{6 m (m - 4) \cancel{(m + 4)}}{\cancel{(m + 4)} 18 (m + 9)}
\][/tex]
Divide [tex]\(6 m\)[/tex] by [tex]\(18\)[/tex]:
[tex]\[
\frac{m (m - 4)}{3 (m + 9)}
\][/tex]
This can be simplified further:
[tex]\[
\frac{m^2 - 4m}{3 (m + 9)}
\][/tex]
The simplified expression is:
[tex]\[
\boxed{\frac{m^2 - 4m}{3 (m + 9)}}
\][/tex]
