1. **State the problem:** Simplify the expression $$(2x+3y)^2-(x-4y)^2$$. 2. **Recall the formula:** This expression is a difference of squares, which follows the identity $$a^2 - b^2 = (a-b)(a+b)$$. 3. **Apply the formula:** Let $$a = 2x+3y$$ and $$b = x-4y$$, so $$ (2x+3y)^2-(x-4y)^2 = ((2x+3y)-(x-4y))((2x+3y)+(x-4y)) $$ 4. **Simplify each factor:** - First factor: $$ (2x+3y)-(x-4y) = 2x + 3y - x + 4y = (2x - x) + (3y + 4y) = x + 7y $$ - Second factor: $$ (2x+3y)+(x-4y) = 2x + 3y + x - 4y = (2x + x) + (3y - 4y) = 3x - y $$ 5. **Write the final simplified expression:** $$ (2x+3y)^2-(x-4y)^2 = (x + 7y)(3x - y) $$ This is the fully simplified form of the original expression.