The variables $x$ and $y$ satisfy $f(x,y) = x^5 + xy^4 + y + x^2$. In a neighbourhood of (0,0), $y$ is approximately
(i) $1+x+x^2$
(ii) $x+2x^2$
(iii) $-2x^2$
(iv) $-x^2$
I don't know how to proceed using implicit function theorem. I found that $f(y) = 1$, not 0 hence there must exist a neighborhood such that $y=g(x)$ and I can rewrite $f(x,y)=0$ as $f(x,g(x))=0$ and I know the formula for computing derivatives: $f'(x,y)= \frac{-f'_x(x,y)}{f'_y(x,y)}$ but I don't get how I am supposed to use that to solve the problem.
If I were you I would find first $m:=\displaystyle\left.\frac{dy}{dx}\right|_{(0,0)}$ and then $a:=\displaystyle\left.\frac{d^2y}{dx^2}\right|_{(0,0)}$.
Then $$y(x)\approx \frac{a}{2}x^2+mx+0.$$
This is the second order Taylor Series approximation.