For which real numbers $c$ is there a straight line that intersects the curve $y = x^4 + 9x^3 + c x^2 + 9x + 4$ in four distinct points?

Question

For which real numbers $c$ is there a straight line that intersects the curve $y = x^4 + 9x^3 + c x^2 + 9x + 4$ in four distinct points?

I don't quite the understand the solution which is in terms of taking the second derivative. Can some one explain it? Here is the link to the solution: http://www.cs.cornell.edu/~asdas/imo/putnam/psoln/psol948.html