1. **State the problem:** We want to express the rational function $$\frac{x^4+3x^3+4x^2+3x+1}{x(x^2+1)^2}$$ as a sum of partial fractions. 2. **Set up the partial fraction decomposition:** Since the denominator is $$x(x^2+1)^2$$, the decomposition form is: $$\frac{A}{x} + \frac{Bx + C}{x^2+1} + \frac{Dx + E}{(x^2+1)^2}$$ 3. **Write the equation:** $$x^4+3x^3+4x^2+3x+1 = A(x^2+1)^2 + (Bx + C)x(x^2+1) + (Dx + E)x$$ 4. **Expand each term:** - Expand $$A(x^2+1)^2 = A(x^4 + 2x^2 + 1) = Ax^4 + 2Ax^2 + A$$ - Expand $$(Bx + C)x(x^2+1) = (Bx + C)(x^3 + x) = Bx^4 + Bx^2 + Cx^3 + Cx$$ - Expand $$(Dx + E)x = Dx^2 + Ex$$ 5. **Combine all terms:** $$Ax^4 + 2Ax^2 + A + Bx^4 + Bx^2 + Cx^3 + Cx + Dx^2 + Ex$$ Group by powers of $$x$$: $$ (A + B)x^4 + Cx^3 + (2A + B + D)x^2 + (C + E)x + A $$ 6. **Match coefficients with the numerator:** From $$x^4 + 3x^3 + 4x^2 + 3x + 1$$, equate coefficients: - $$x^4: A + B = 1$$ - $$x^3: C = 3$$ - $$x^2: 2A + B + D = 4$$ - $$x^1: C + E = 3$$ - Constant: $$A = 1$$ 7. **Solve the system:** - From constant: $$A = 1$$ - From $$x^4$$: $$1 + B = 1 \Rightarrow B = 0$$ - From $$x^3$$: $$C = 3$$ - From $$x^1$$: $$3 + E = 3 \Rightarrow E = 0$$ - From $$x^2$$: $$2(1) + 0 + D = 4 \Rightarrow 2 + D = 4 \Rightarrow D = 2$$ 8. **Write the final partial fraction decomposition:** $$\frac{1}{x} + \frac{0x + 3}{x^2 + 1} + \frac{2x + 0}{(x^2 + 1)^2} = \frac{1}{x} + \frac{3}{x^2 + 1} + \frac{2x}{(x^2 + 1)^2}$$