1. **State the problem:** Simplify the expression $$(x+4)(3x+11) - (x+4)(6x+7).$$ 2. **Formula and rule:** Use the distributive property: $$a(b+c) = ab + ac$$ and factor common terms. 3. **Step 1:** Factor out the common term $(x+4)$: $$ (x+4)(3x+11) - (x+4)(6x+7) = (x+4)\big[(3x+11) - (6x+7)\].$$ 4. **Step 2:** Simplify inside the brackets: $$ (3x+11) - (6x+7) = 3x + 11 - 6x - 7 = (3x - 6x) + (11 - 7) = -3x + 4.$$ 5. **Step 3:** Substitute back: $$ (x+4)(-3x + 4).$$ 6. **Step 4:** Expand the product: $$ (x+4)(-3x + 4) = x(-3x) + x(4) + 4(-3x) + 4(4) = -3x^2 + 4x - 12x + 16.$$ 7. **Step 5:** Combine like terms: $$ -3x^2 + (4x - 12x) + 16 = -3x^2 - 8x + 16.$$ **Final answer:** $$\boxed{-3x^2 - 8x + 16}.$$