# Solving system of equations to find critical points of multivariable function

I've been given the following function and asked to find all critical points.

$$f(x,y)=y^3+2x^2y+5xy$$

I began by finding partial derivatives $$\frac{\partial f}{\partial x}=4xy+5y$$ $$\frac{\partial f}{\partial y}=3y^2+2x^2+5x$$

I understand that to find critical points I need to equate both partial derivatives to $$0$$ and solve the system but I'm quite sure how to solve this particular system. Any tips?

Hint: $$4xy+5y=0 \rightarrow y=0$$ or $$x=-\frac{5}{4}$$. Now, consider these two cases and solve the second equation substituting $$y$$ or $$x$$. If $$y=0$$ you get $$5x=0$$ and so forth.
The first equation gives $$y(4x+5)=0.$$ Can you end it now?
I got: $$\left\{(0,0),(-2.5,0),\left(-\frac{5}{4},\frac{5}{2\sqrt6}\right), \left(-\frac{5}{4},-\frac{5}{2\sqrt6}\right)\right\}.$$