Algebra

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(a) Let's start by squaring. We have $a^2+b^2+2ab=16.$ Thus $a^2+b^2+2ab=a^4+b^4.$We soon see that the only solutions for $\boxed{ab=0.}$

(b) Problem explictly states that $a+b=4$.

(c) We know that $a+b=4$ and $ab=0$. Thus one of either $a$ or $b$ must be zero.

There are four solutions: $(a,b)=(2,0);(-2,0);(0,2);(0,-2).$