The number of points on the curve  $\mathrm{y}=54{\mathrm{x}}^5-135{\mathrm{x}}^4-70{\mathrm{x}}^3+180{\mathrm{x}}^2+210\mathrm{x}$ at which the normal lines are parallel to $\mathrm{x}+90\mathrm{y}+2=0$ is:

$\frac{\mathrm{dy}}{\mathrm{dx}}=270{\mathrm{x}}^4-540{\mathrm{x}}^3-210{\mathrm{x}}^2+360\mathrm{x}+210$
Slope of normal =-1/90

Slope of tangent=90 

$270{\mathrm{x}}^4-540{\mathrm{x}}^3-210{\mathrm{x}}^2+360\mathrm{x}+210=90$

$9{\mathrm{x}}^4-18{\mathrm{x}}^3-7{\mathrm{x}}^2+12\mathrm{x}+4=0$

$\mathrm{x}=1,2\text{ are roots}$
$\begin{array}{l}1\overline{)\begin{array}{l}9 -18 -7 12 4\\ 0 9 -9 -16 -4\end{array}}\\ 2\overline{)\begin{array}{l}9 -9 -16 -4 \overline{)0}\\ 0 18 18 4\end{array}}\\ 9 9 2 \overline{)0}\\ 9{\mathrm{x}}^2+9\mathrm{x}+2=0\\ \mathrm{\Delta }=81-4.9.2>0\end{array}$
Four roots=4such normals