Find the value(s) of x for which F(x) = g(x)

I suppose you mean F(x) = ∫f(x)dx

F(x) = ∫f(x)dx = ∫(x^4 - 2x^2)dx = (x^5)/5 - (2x^3)/3

Solve F(x) = g(x) for x:
F(x) = g(x) = (x^5)/5 - (2x^3)/3 = 2x^2 [-2x^2]
(x^5)/5 - (2x^3)/3 - 2x^2 = 2x^2 * (x^3/10 - x/3 - 1) = 0

So we get
x_0 = 0
x_1 = 0

For the other values you could use methods like completing the square or just the polynomial division.
[input]solve((x^3)/10 - x/3 - 1 = 0)[/input]

I don't think (s)he was talking about integrals.

f(x)=x^4-2x^2 = g(x)=2x^2

x^4-2x^2=2x^2

First, add 2x^2 to each side:

x^4=4x^2

Then divide by x^2

x^2=4

Then take the square root:

x=+-sqrt(4)=+-2

So there are two values for which f(x)=g(x) and those are x=2 and x=-2. Put those numbers into the equations to verify your answer