Find the asymptotes of the curve
$\displaystyle
x^3 + 4x^2y + 4xy^2 +5x^2+15xy+10y^2-2y+1=0
$
I would rearrange the equation 1st
$\displaystyle y^2(4x+10) +y(4x^2 +15x -2) +(x^3+5x^2+1)$
and then use the quad equation.
When you do all the algebra work you end up with
$\displaystyle y = \frac{-4x^2-15x+2 ^+_- \sqrt{9x^2-76x-36}}{8x+20}$
So your vertical asymptote is
$\displaystyle x = \frac{-20}{8}$
and your slant asymptote is
$\displaystyle y= -\frac{x}{2} - \frac{5}{8}$
and
$\displaystyle 9x^2 - 76x -36 \geq 0$