# Math Help - a confusing complex fraction

1. ## a confusing complex fraction

here's the complex fraction:

By the way, my prof. says that the answer is -3m

2. We have to simplify the expression:

$\displaystyle\frac{\displaystyle\frac{5m-6}{2}-\frac{18m^2-9m-20}{12m+10}}{\displaystyle\frac{1+m(3n^2-1)}{2-\displaystyle\frac{3m^2-8m+4}{2-m}}-n^2}$

We evaluate the numerator:

$\frac{5m-6}{2}-\frac{18m^2-9m-20}{2(6m+5)}=\frac{(5m-6)(6m+5)-18m^2+9m+20}{2(6m+5)}=$

$\frac{12m^2-2m-10}{2(6m+5)}=\frac{2(m-1)(6m+5)}{2(6m+5)}=m-1$

Now we evaluate the denominator:

$\frac{1+m(3n^2-1)}{2-\displaystyle\frac{3m^2-8m+4}{2-m}}-n^2=\frac{2-m+(2m-m^2)(3n^2-1)}{4-2m-3m^2+8m-4}-n^2=$

$=\frac{m^2-3m+2}{-3m(m-2)}=\frac{m-1}{-3m}$

Now, $\frac{m-1}{\displaystyle\frac{m-1}{-3m}}=(m-1)\cdot\frac{-3m}{m-1}=-3m$