1. **State the problem:** Divide the polynomial $2x^2 + 11x + 5$ by the binomial $2x + 1$. 2. **Formula and rules:** Polynomial division is similar to long division with numbers. We divide the leading term of the dividend by the leading term of the divisor, multiply the divisor by that result, subtract, and repeat. 3. **Step 1:** Divide the leading term $2x^2$ by $2x$: $$\frac{2x^2}{2x} = x$$ 4. **Step 2:** Multiply the divisor by $x$: $$x \times (2x + 1) = 2x^2 + x$$ 5. **Step 3:** Subtract this from the original polynomial: $$\left(2x^2 + 11x + 5\right) - \left(2x^2 + x\right) = (2x^2 - 2x^2) + (11x - x) + 5 = 0 + 10x + 5 = 10x + 5$$ 6. **Step 4:** Divide the new leading term $10x$ by $2x$: $$\frac{10x}{2x} = 5$$ 7. **Step 5:** Multiply the divisor by $5$: $$5 \times (2x + 1) = 10x + 5$$ 8. **Step 6:** Subtract this from the remainder: $$\left(10x + 5\right) - \left(10x + 5\right) = 0$$ 9. **Result:** The quotient is $x + 5$ and the remainder is $0$, so $$\frac{2x^2 + 11x + 5}{2x + 1} = x + 5$$