<s>[INST] 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
 [/INST] The tangent line equation at $x=p$ and $x=q$ is:
$$y=p^4+4p^3-16p^2+6p-5+(4p^3+12p^2-32p+6)(x-p);\\
y=q^4+4q^3-16q^2+6q-5+(4q^3+12q^2-32q+6)(x-q).\\$$
By equating the coefficients, simplifying and denoting $a=p+q, b=pq$ we get:
$$\begin{cases}b=a^2+3a-8\\
3a^3-(6a+8)b+8a^2-16a=0\end{cases}$$
Solving the system we get:
$$a=-2,b=-10=pq.$$
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