Simplfying expression efficiently What would be the best way to simplify such an expression? Simply foiling out the expression? could I take out $\frac{b-a}{4}$ and simplifying?
$$\frac{3}{4}(b-a)\left( \frac{2a}{3} +\frac
{b}{3} \right)^2 + \left( \frac{b-a}{4} \right)b^2$$ I know that this is equivalent to $\frac{b^3-a^3}{3}$

Well, first we notice that there's a common factor of $\frac{b - a}{4}$ in there, so we'll pull that out, leaving 
$$\left(\frac{2a+b}{3}\right)^2 + b^2.$$
Expanding that bracket out, we obtain 
$$\frac{4a^2 + 4ab + b^2}{3} + b^2$$
Combining those fractions, that's 
$$\frac{4a^2+4ab+4b^2}{3}$$
Pulling out the $\frac{4}{3}$ (which, in particular, cancels out the factor of $\frac{1}{4}$ that we pulled out in the first step, so our pulled-out bit is now $\frac{b-a}{3}$), we're left with
$a^2 + ab + b^2.$
At this point, there's not much left to do besides combining the two parts again, giving our full expression as being equal to 
$$\frac{(b-a)(a^2+ab+b^2)}{3}.$$
Multiplying through that top thing, we obtain 
$$\frac{b^3-a^3}{3},$$ 
as expected. 
