1. We are asked to factorize the expression $$(2x + 9)^2 - (x + 4)^2$$. 2. Recognize this as a difference of squares, which has the formula $$a^2 - b^2 = (a - b)(a + b)$$. 3. Let $$a = 2x + 9$$ and $$b = x + 4$$. 4. Then the expression can be written as $$(2x + 9 - (x + 4))(2x + 9 + (x + 4))$$. 5. Simplify each factor: - First factor: $$2x + 9 - x - 4 = (2x - x) + (9 - 4) = x + 5$$ - Second factor: $$2x + 9 + x + 4 = (2x + x) + (9 + 4) = 3x + 13$$ 6. Therefore, the fully factorized form is $$ (x + 5)(3x + 13) $$. Final answer: $$ (x + 5)(3x + 13) $$.