First of all you need to solve for where the two curves intersect.

Solve simultaneously:

f(x) = x^3 + x^2 + 1 and g(x) = -x^2 + 3x +1

x^3 + 2x^2 - 3x = 0

x (x^2 + 2x - 3) = 0

x (x + 3)(x - 1) = 0

so the two curves actually intersect at 3 spots... at

x = -3, y = -17

x = 0, y = 1

and x = 1, y = 3

Between -3 and 0, we see that f(x) > g(x) and between 0 and 1, we see that g(x) > f(x)

Therefore, the the area bounded by the two curves should be given by the formula:

integral from -3 to 0 of [f(x) - g(x)] + integral from 0 to 1 of [g(x) - f(x)]

:-)

Hope that helps... and sorry about the notation... the math tags don't seem to be working...