This problem presents us with the derivative of the profit function, P, with respect to time, t.
To calculate the profit function itself we must find the integral of this derivative. This does not appear to be very simple since the derivative is identified as the product of these two expressions: (3t^2 + 2) and (t^3 + 2t + 2) raised to the power of the fraction 1/3. However, if you define u = t^3 + 2t + 2 then du/dt = 3t^2 + 2 which is the first expression.
This means that we may rewrite the original integral as u raised to the power of the fraction 1/3.
Now, if you find the antiderivative of u^(1/3) we get u^(4/3) divided by 4/3. (Note: recall that the integration of x^n is defined as x^(n+1) divided by (n+1). This is the same thing with n = 1/3)
Dividing by 4/3 is equivalent to multiplying by 3/4, so we may rewrite our answer as 3/4 u^(4/3). By substituting u with its equivalent in terms of t we have P(t) = 3/4 (t^3 + 2t + 2)^4/3 + some arbitrary constant.
If you want to know what is the profit when t=4, change t to 4 and compute 3/4 (4^3 + 2(4) + 2)^(4/3) = 3/4(64 + 8 + 2)^(4/3) = 3/4 (74)^(4/3) = approximately 233. I did not consider what the value might be for the arbitrary constant. I believe it is a fair assumption to make it zero.