Find dy/dx by implicit differentiation x^3+3x^2y+y^3=8

Just remember, for y, just leave the dy/dx there

3x²+3(2xy+x²dy/dx)+3y²dy/dx=0
3x²+6xy+3x²dy/dx+3y²dy/dx=0
minus (3x²+6xy) from both sides
3x²dy/dx+3y²dy/dx=-3x²-6xy
undistribute dy/dx
dy/dx(3x²+3y²)=-3x-3xy
divide both sides by (3x²+3y²)

First subtract 10 from both side
y^2+6y-16=0
(y+8)(y-2)=0
-8 and 2