1. **State the problem:** Express the rational function $$\frac{x^2 + 4}{x^3 + 2x}$$ in partial fractions. 2. **Factor the denominator:** $$x^3 + 2x = x(x^2 + 2)$$ 3. **Set up the partial fractions:** Since $x$ is linear and $x^2 + 2$ is irreducible quadratic, write: $$\frac{x^2 + 4}{x(x^2 + 2)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 2}$$ 4. **Multiply both sides by the denominator to clear fractions:** $$x^2 + 4 = A(x^2 + 2) + (Bx + C)(x)$$ 5. **Expand the right side:** $$x^2 + 4 = A x^2 + 2A + B x^2 + C x$$ 6. **Group like terms:** $$x^2 + 4 = (A + B) x^2 + C x + 2A$$ 7. **Equate coefficients of powers of $x$ on both sides:** - Coefficient of $x^2$: $1 = A + B$ - Coefficient of $x$: $0 = C$ - Constant term: $4 = 2A$ 8. **Solve the system:** From $4 = 2A$, we get $A = 2$. From $0 = C$, we get $C = 0$. From $1 = A + B$, substitute $A=2$: $$1 = 2 + B \implies B = 1 - 2 = -1$$ 9. **Write the final partial fraction decomposition:** $$\frac{x^2 + 4}{x^3 + 2x} = \frac{2}{x} + \frac{-x}{x^2 + 2}$$ This is the expression in partial fractions.