Tangent in (at least) two points

Question

Let $$y=x^4-2x^3-3^2+5x+6$$

Find a line that's a tangent on $y$ in at least two points.

The only idea I've had is to apply $y-y_0 = y\prime (x_0) \cdot (x-x_0)$, but that didn't get me far.

Any hints would be appreciated.

Answer 1

Hint: Let $y=mx+c$ be the tangent. Then the equation in $x$

$$x^4-2x^3-3x^2+5x+6-(mx+c)=0$$

has two double roots. Therefore,

$$x^4-2x^3-3x^2+(5-m)x+(6-c)=(x-\alpha)^2(x-\beta)^2$$ for some $\alpha,\beta\in\mathbf{R}$.

By comparing coefficients, $m$ and $c$ can be determined.