What's the best way to compute $\frac{a^4 + b^4 + c^4}{a^2 + b^2 + c^2}$ So, my teacher gave us this to compute yesterday, and I'm completly confused on how should I proceed : 
$$\frac{1^4 + 2012^4 +2013^4}{1^2 + 2012^2 + 2013^2}$$
I've tried several ways, but most of them are very long, for example I've simplified both numbers :
$2012^2 = (2 * 10^3)^2 + 12^2 + 24 * 10^3$ 
$2013^2 = (2 * 10^3)^2 + 13^2 + 26 * 10^3$
I can't see how this could help me solve this problem . So how how should I go with this kind of problems in general ?

Hint The given triple $(1, 2012, 2013)$ has the special feature that $1 + 2012 = 2013$. This motivates writing the quotient as
$$\frac{a^4 + b^4 + (a + b)^4}{a^2 + b^2 + (a + b)^2}$$
for $a = 1, b = 2012$.

Additional hint Expanding the numerator, we can see that it factors as $2(a^2+ab+b^2)^2$

