1. **Problem Statement:** Compute the integral $$\int \frac{4x^3 + 2x}{(x^4 + x^2 + 5)^5} \, dx$$. 2. **Formula and Approach:** We use substitution for integrals of the form $$\int \frac{f'(x)}{(f(x))^n} \, dx$$. Let $$u = x^4 + x^2 + 5$$. Then, $$\frac{du}{dx} = 4x^3 + 2x$$. 3. **Substitution:** Rewrite the integral in terms of $$u$$: $$\int \frac{4x^3 + 2x}{(x^4 + x^2 + 5)^5} \, dx = \int \frac{du}{u^5}$$. 4. **Integration:** $$\int u^{-5} \, du = \frac{u^{-4}}{-4} + C = -\frac{1}{4u^4} + C$$. 5. **Back-substitute:** $$-\frac{1}{4(x^4 + x^2 + 5)^4} + C$$. **Final answer:** $$\boxed{-\frac{1}{4(x^4 + x^2 + 5)^4} + C}$$