Final answer:
To simplify the given expression \(\frac{8m^2+16m}{m-3} \cdot \frac{m^2-m-6}{4m^3+8m^2}\), factorize the polynomials, cancel the common factors, and simplify to get \(\frac{2}{m}\).
Step-by-step explanation:
The student has asked for the simplification of the expression \(\frac{8m^2+16m}{m-3} \cdot \frac{m^2-m-6}{4m^3+8m^2}\). To simplify this expression, we will factorize the polynomials in the numerators and denominators, and then cancel common factors.
First, we factorize 8m^2+16m as 8m(m+2). Next, m^2-m-6 can be factorized as (m-3)(m+2). The denominator 4m^3+8m^2 can be factorized as 4m^2(m+2). After factorizing, our expression becomes:
\(\frac{8m(m+2)}{m-3} \cdot \frac{(m-3)(m+2)}{4m^2(m+2)}\)
Now, we cancel the common factors (m+2) and (m-3) across the numerators and denominators, which leaves:
\(\frac{8m}{4m^2}\)
Finally, we simplify this by canceling the m and dividing 8 by 4, resulting in:
\(\frac{2}{m}\)