If you have given the question exactly, this is not a hard problem. As you have given it, it does not ask you to find the roots of the given cubic, but rather to show that it has three distinct real roots in a given interval.
You need to know that a cubic polynomial has AT LEAST one distinct real root and AT MOST three distinct real roots over the entire domain of real numbers.
Calculate the value of your cubic at -4, -3, -2, -1, 0, 1, 2, 3, 4.
Do you see from that there must be at least one root between 0 and 1? What other pairs of numbers must have at least one root between them?
Is it clear now or does this sound like mumbo jumbo?
