1. State the problem: We are asked to decompose the expression $\frac{1}{2x^2 + x}$ into partial fractions. 2. Factor the denominator: The denominator $2x^2 + x$ can be factored as $x(2x + 1)$. 3. Set up the partial fraction decomposition: $$\frac{1}{x(2x+1)} = \frac{A}{x} + \frac{B}{2x+1}$$ where $A$ and $B$ are constants to be determined. 4. Multiply both sides by the denominator $x(2x + 1)$ to clear the fractions: $$1 = A(2x + 1) + Bx$$ 5. Expand the right side: $$1 = 2Ax + A + Bx = (2A + B)x + A$$ 6. Equate coefficients of like terms from both sides: For the constant terms: $A = 1$ For the $x$ terms: $2A + B = 0$ 7. Substitute $A = 1$ into $2A + B = 0$: $$2(1) + B = 0 \implies 2 + B = 0 \implies B = -2$$ 8. Write the final partial fraction decomposition: $$\frac{1}{2x^2 + x} = \frac{1}{x} - \frac{2}{2x + 1}$$