1. The problem is to solve the equation $$(x + 9)^2 - (x + 8)^2 = (2x + 11)^2.$$\n\n2. Use the formula for the square of a binomial: $$(a \pm b)^2 = a^2 \pm 2ab + b^2.$$\n\n3. Expand each square:\n$$(x + 9)^2 = x^2 + 18x + 81,$$\n$$(x + 8)^2 = x^2 + 16x + 64,$$\n$$(2x + 11)^2 = 4x^2 + 44x + 121.$$\n\n4. Substitute these into the original equation:\n$$x^2 + 18x + 81 - (x^2 + 16x + 64) = 4x^2 + 44x + 121.$$\n\n5. Simplify the left side by distributing the minus sign:\n$$x^2 + 18x + 81 - x^2 - 16x - 64 = 4x^2 + 44x + 121.$$\n\n6. Cancel $\cancel{x^2}$ and combine like terms:\n$$2x + 17 = 4x^2 + 44x + 121.$$\n\n7. Bring all terms to one side to set the equation to zero:\n$$2x + 17 - 4x^2 - 44x - 121 = 0,$$\nwhich simplifies to\n$$-4x^2 - 42x - 104 = 0.$$\n\n8. Multiply both sides by $-1$ to simplify:\n$$\cancel{-}4x^2 - 42x - 104 = 0 \Rightarrow 4x^2 + 42x + 104 = 0.$$\n\n9. Divide the entire equation by 2 to simplify coefficients:\n$$\frac{4x^2}{2} + \frac{42x}{2} + \frac{104}{2} = 0 \Rightarrow 2x^2 + 21x + 52 = 0.$$\n\n10. Identify coefficients for the quadratic formula:\n$$a = 2, \quad b = 21, \quad c = 52.$$\n\n11. Use the quadratic formula to solve for $x$:\n$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-21 \pm \sqrt{21^2 - 4 \cdot 2 \cdot 52}}{2 \cdot 2}.$$\n\n12. Calculate the discriminant:\n$$21^2 - 4 \cdot 2 \cdot 52 = 441 - 416 = 25.$$\n\n13. Calculate the roots:\n$$x = \frac{-21 \pm 5}{4}.$$\n\n14. Find each solution:\n$$x_1 = \frac{-21 + 5}{4} = \frac{-16}{4} = -4,$$\n$$x_2 = \frac{-21 - 5}{4} = \frac{-26}{4} = -6.5.$$\n\n15. Therefore, the solutions to the equation are $$x = -4$$ and $$x = -6.5.$$