Rank and nullity of projection onto one vector

For a non-zero $v\in\mathbb{R}^n$, let $P_v=\dfrac{vv^T}{\|v\|^2}$ be the projection onto the line through $v$. Let $r$ be its rank and $m$ its nullity. The quantity $r^2+m^2$ is a polynomial $an^2+bn+c$ with integer coefficients; give $a+b+c$.

$1$. Every output $P_vx$ is a multiple of $v$ and $P_vv=v$, so the rank is $1$ and, by rank-nullity, the nullity is $n-1$. Then $1+(n-1)^2=n^2-2n+2$, so $a+b+c=1-2+2=1$.