Question

Determine the equation of the polynomial, p(x),in both factored and standard form.

Answer 1

We can see that the graph touches $x=-1$ without crossing the x-axis (i.e. it is a double solution), and then there's another zero at $x=2$ (this time it's a crossing zero, so a single solution).

This leads, up to multiple, to the polynomial

$p(x)=a(x+1)^2(x-2)$

If we impose the passing through $(0,4)$ we have

$p(0)=4=a(1)(-2) \iff -2a=4 \iff a=-2$

So, the polynomial is

$p(x)=-2(x+1)^2(x-2)=-2 x^3 + 6 x + 4$

Finally, to solve $p(x)$, simply look at the graph, searching for the points, where the graph is below the x-axis. You can see that this happens only if $x>2$, so that's the solution to your question.