Given our function f(x) = x^3 + 2x^2 - 5x - 6 the shaded region required is the sum of two areas which we will call A (shaded region above the x axis) and B (shaded region below the x axis).

To find A we need to integrate f(x) with respect to x between two limits i.e. from the lowest value of x for region A, to the greatest value of x for region A.

To find these limits we must calculate the values of x at which f(x) cuts across the x axis, and as you know, these are the roots of our cubic equation.

There are a number of ways to find the roots of our equation - the easiest approach in this case seems to be by factorising the cubic.

To factorise f(x) we can use the polynomial factor theorem which tells us that:

if f(a) = 0 then (x-a) is a factor of f(x).

From your graph you might consider the value: a = -1

f(-1) = (-1)^3 + 2(-1)^2 -5(-1) - 6
= 0

This satisfies the condition of the factor theorem so:
=> (x - (-1)) is a factor of f(x).

We can use this to factorise f(x) completely and so find all the roots:

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