Let's multiply the given rational expressions step-by-step.
The problem we have is:
[tex]\[
\frac{7 m^3}{m^2 - m - 6} \cdot \frac{m^2 - 4}{35 m}
\][/tex]
First, we'll factor each part where possible to simplify the expression.
1. Factor [tex]\( m^2 - m - 6 \)[/tex]:
[tex]\[
m^2 - m - 6 = (m - 3)(m + 2)
\][/tex]
Thus:
[tex]\[
\frac{7 m^3}{m^2 - m - 6} = \frac{7 m^3}{(m - 3)(m + 2)}
\][/tex]
2. Factor [tex]\( m^2 - 4 \)[/tex]:
[tex]\[
m^2 - 4 = (m - 2)(m + 2)
\][/tex]
So,
[tex]\[
\frac{m^2 - 4}{35 m} = \frac{(m - 2)(m + 2)}{35 m}
\][/tex]
Next, we multiply the fractions:
[tex]\[
\frac{7 m^3}{(m - 3)(m + 2)} \cdot \frac{(m - 2)(m + 2)}{35 m}
\][/tex]
Now, combine the numerators and denominators:
[tex]\[
\frac{7 m^3 \cdot (m - 2)(m + 2)}{(m - 3)(m + 2) \cdot 35 m}
\][/tex]
We can cancel out the common factors in the numerator and denominator:
- [tex]\( m + 2 \)[/tex] in the numerator and denominator both cancel:
[tex]\[
\frac{7 m^3 \cdot (m - 2)}{(m - 3) \cdot 35 m}
\][/tex]
- [tex]\( 7 m \)[/tex] in the numerator and denominator cancel:
[tex]\[
\frac{7 m^2 \cdot (m - 2)}{(m - 3) \cdot 35}
\][/tex]
This simplifies further to:
[tex]\[
\frac{m^2 \cdot (m - 2)}{5 \cdot (m - 3)}
\][/tex]
So the simplified expression is:
[tex]\[
\frac{m^2 (m - 2)}{5 (m - 3)}
\][/tex]
