1. **State the problem:** We need to find the indefinite integral $$\int 15\sqrt{4 + x} \, dx$$ and express the answer in the form $$a(4 + x)^b + C$$ where $a$ and $b$ are constants to be determined. 2. **Rewrite the integral:** Recall that $$\sqrt{4 + x} = (4 + x)^{\frac{1}{2}}$$ so the integral becomes: $$\int 15 (4 + x)^{\frac{1}{2}} \, dx$$ 3. **Use the power rule for integration:** For any $n \neq -1$, $$\int (x + c)^n \, dx = \frac{(x + c)^{n+1}}{n+1} + C$$ 4. **Apply the formula:** Here, $n = \frac{1}{2}$, so $$\int 15 (4 + x)^{\frac{1}{2}} \, dx = 15 \int (4 + x)^{\frac{1}{2}} \, dx = 15 \cdot \frac{(4 + x)^{\frac{3}{2}}}{\frac{3}{2}} + C$$ 5. **Simplify the fraction:** $$15 \cdot \frac{(4 + x)^{\frac{3}{2}}}{\frac{3}{2}} = 15 \cdot (4 + x)^{\frac{3}{2}} \cdot \frac{2}{3}$$ 6. **Cancel common factors:** $$15 \cdot \frac{2}{3} = \cancel{15} \cdot \frac{2}{\cancel{3}} = 5 \cdot 2 = 10$$ 7. **Final answer:** $$\int 15\sqrt{4 + x} \, dx = 10 (4 + x)^{\frac{3}{2}} + C$$ Thus, the constants are $a = 10$ and $b = \frac{3}{2}$.