# How do you multiply (6a^2-30a)/(a-2) * (a^2+2a-8)/(2a^3-10a^2)?

May 11, 2015

The answer is : $\frac{3 a + 12}{a}$

Firstly, let's factor all the polynomials :

$\frac{6 {a}^{2} - 30 a}{a - 2} \cdot \frac{{a}^{2} + 2 a - 8}{2 {a}^{3} - 10 {a}^{2}} = \frac{6 a \left(a - 5\right)}{a - 2} \cdot \frac{\left(a + 4\right) \left(a - 2\right)}{2 {a}^{2} \left(a - 5\right)}$.

Now you can multiply the numerators together and then the denominators :

$\frac{6 a \left(a - 5\right)}{a - 2} \cdot \frac{\left(a + 4\right) \left(a - 2\right)}{2 {a}^{2} \left(a - 5\right)} = \frac{6 a \left(a - 5\right) \left(a + 4\right) \left(a - 2\right)}{2 {a}^{2} \left(a - 2\right) \left(a - 5\right)}$

Since the numerator and the denominator both have $a$, $\left(a - 5\right)$ and $\left(a - 2\right)$ in common, you can factor them out :

$\frac{6 a \left(a - 5\right) \left(a + 4\right) \left(a - 2\right)}{2 {a}^{2} \left(a - 2\right) \left(a - 5\right)} = \frac{6 \cdot \left(a + 4\right)}{2 a}$

You can now divide 6 by 2 and calculate the answer :

$\frac{6 \cdot \left(a + 4\right)}{2 a} = \frac{3 \cdot \left(a + 4\right)}{a} = \frac{3 a + 12}{a} = 3 + \frac{12}{a}$