A common tangent line 
The graph of $f(x)=x^4+4x^3-16x^2+6x-5$ has a common tangent line at $x=p$ and $x=q$. Compute the product $pq$.

So what I did is I took the derivative and found out that $p^2+3p+q^2+3q+pq=0$. However when I tried to factorize it I didn't find out an obvious solution. Can someone hint me what to do next? Thanks in advance

Let the equation of the common tangent be $y=mx+b$.
Solving simultaneously the equations of the common tangent and $f(x)$, we get
$x^4+4x^3-16x^2+(6-m)x-(b+5)=0$
This equation will have two double roots (since a line is touching a curve twice), so let the roots be $p,\,p,\,q,\,q$.
Can you take it from here?
