1. **State the problem:** We need to calculate the payoff of the insurance claim \( C = 5E[X^3] + 6E[X^2] \) where \(X\) is a gamma distributed random variable with parameters \(\alpha = 3\) and \(\beta = 4\).\n\n2. **Recall properties of the gamma distribution:** For a gamma distribution with shape \(\alpha\) and scale \(\beta\), the \(n\)-th moment is given by \(E[X^n] = \beta^n \frac{\Gamma(\alpha + n)}{\Gamma(\alpha)}\).\n\n3. **Calculate \(E[X^2]\):**\n$$E[X^2] = \beta^2 \frac{\Gamma(\alpha + 2)}{\Gamma(\alpha)} = 4^2 \frac{\Gamma(3 + 2)}{\Gamma(3)} = 16 \frac{\Gamma(5)}{\Gamma(3)}$$\nRecall \(\Gamma(n) = (n-1)!\) for integer \(n\), so \(\Gamma(5) = 4! = 24\) and \(\Gamma(3) = 2! = 2\).\nThus,\n$$E[X^2] = 16 \times \frac{24}{2} = 16 \times 12 = 192$$\n\n4. **Calculate \(E[X^3]\):**\n$$E[X^3] = \beta^3 \frac{\Gamma(\alpha + 3)}{\Gamma(\alpha)} = 4^3 \frac{\Gamma(3 + 3)}{\Gamma(3)} = 64 \frac{\Gamma(6)}{\Gamma(3)}$$\nUsing factorials, \(\Gamma(6) = 5! = 120\) and \(\Gamma(3) = 2\).\nSo,\n$$E[X^3] = 64 \times \frac{120}{2} = 64 \times 60 = 3840$$\n\n5. **Calculate the payoff \(C\):**\n$$C = 5E[X^3] + 6E[X^2] = 5 \times 3840 + 6 \times 192 = 19200 + 1152 = 20352$$\n\n**Final answer:** \(C = 20352\)