Intersection of curve $ y = x^4 – 6x^3 + 12x^2 + cx + 1$ The number of integers in the range of 'c' such that there exists a line which intersects the curve   $ y = x^4 – 6x^3 + 12x^2 + cx + 1$ at four distinct points. 
My approach we need to intersect with line $y=mx+C$
Substituting we get  $x^4 – 6x^3 + 12x^2 + cx + 1-mx-C=0$ 
Now this is an equation of polynomial of degree 4
$x^4 – 6x^3 + 12x^2 + (c-m)x + 1-C=0$ 
All four roots needs to be real.
I don't have any idea how to check whether all the roots are real or not.

The second derivative $y''(x)=12x^2-36x+24=12(x-1)(x-2)$ has roots $\{1,2\}$ independently of $c$. Thus, $y(x)$ has two points of inflection at $(1, 8+c)$ and $(2, 17+2c)$. A line that goes through these two points will always intersect $y(x)$ at 4 points (two points of indflection and $\{(3\pm\sqrt5)/2\}$).
