1. Stating the problem: We want to decompose the rational expression $$\frac{5x+1}{(2x+1)^2}$$ into partial fractions of the form $$\frac{A}{2x+1} + \frac{B}{(2x+1)^2}$$. 2. Formula and rules: For repeated linear factors like $$(2x+1)^2$$, the partial fraction decomposition includes terms with denominators $$(2x+1)$$ and $$(2x+1)^2$$. 3. Set up the equation: $$\frac{5x+1}{(2x+1)^2} = \frac{A}{2x+1} + \frac{B}{(2x+1)^2}$$ Multiply both sides by $$(2x+1)^2$$ to clear denominators: $$5x+1 = A(2x+1) + B$$ 4. Expand the right side: $$5x+1 = 2Ax + A + B$$ 5. Equate coefficients of like terms: - Coefficient of $x$: $$5 = 2A$$ - Constant term: $$1 = A + B$$ 6. Solve for $A$: $$A = \frac{5}{2}$$ 7. Substitute $A$ into the constant term equation: $$1 = \frac{5}{2} + B$$ 8. Solve for $B$: $$B = 1 - \frac{5}{2} = \frac{2}{2} - \frac{5}{2} = -\frac{3}{2}$$ 9. Final answer: $$\frac{5x+1}{(2x+1)^2} = \frac{\frac{5}{2}}{2x+1} + \frac{-\frac{3}{2}}{(2x+1)^2} = \frac{5/2}{2x+1} - \frac{3/2}{(2x+1)^2}$$