Solution-
General 4th degree polynomial equation with roots as a, b, c, d is,
\(y=(x-a)(x-b)(x-c)(x-d)\)
If the graph touches or bounces on a root, then that root has an even multiplicity.
If the graph goes through a root, then that root has an odd multiplicity.
Here, the roots are -1, 2, 4. (∵ as the graph crosses or touches x-axis or y=0 line)
At x=2, the graph touches the x-axis, so it has a multiplicity of 2
Now, we have to consider the end behavior or leading co-efficient.\(As\ x\rightarrow -\infty,\ f(x)\rightarrow -\infty\)
and
\(As\ x\rightarrow +\infty,\ f(x)\rightarrow -\infty\)
So, the graph must be function with even degree and negative leading co-efficient.
So, the final equation becomes,
\(y=-(x+1)(x-2)^2(x-4)\)
