1. We are asked to evaluate the integral $$\int 4x^2 \sqrt{x^3 - 5} \, dx$$. 2. Let us use substitution. Set $$u = x^3 - 5$$. 3. Then, the derivative is $$du = 3x^2 \, dx$$, so $$x^2 \, dx = \frac{du}{3}$$. 4. Rewrite the integral in terms of $$u$$: $$\int 4x^2 \sqrt{x^3 - 5} \, dx = \int 4 \sqrt{u} \cdot x^2 \, dx = \int 4 \sqrt{u} \cdot \frac{du}{3} = \frac{4}{3} \int u^{1/2} \, du$$. 5. Integrate $$\int u^{1/2} \, du$$ using the power rule: $$\int u^{1/2} \, du = \frac{u^{3/2}}{(3/2)} = \frac{2}{3} u^{3/2}$$. 6. Substitute back to get the final answer: $$\frac{4}{3} \cdot \frac{2}{3} u^{3/2} + C = \frac{8}{9} (x^3 - 5)^{3/2} + C$$.