1. Problem: Factorize the expression $a^2 - 12a - 28 + 16b - b^2$. 2. Formula and rules: To factorize, group terms and look for patterns such as difference of squares, perfect square trinomials, or common factors. 3. Step 1: Rearrange terms grouping $a$ and $b$ terms: $$a^2 - 12a - b^2 + 16b - 28$$ 4. Step 2: Complete the square for $a$ and $b$ terms separately. For $a$ terms: $$a^2 - 12a = a^2 - 12a + 36 - 36 = (a - 6)^2 - 36$$ For $b$ terms: $$-b^2 + 16b = -(b^2 - 16b) = -(b^2 - 16b + 64 - 64) = -(b - 8)^2 + 64$$ 5. Step 3: Substitute back: $$(a - 6)^2 - 36 - (b - 8)^2 + 64 - 28$$ Simplify constants: $$-36 + 64 - 28 = 0$$ So expression becomes: $$(a - 6)^2 - (b - 8)^2$$ 6. Step 4: Recognize difference of squares: $$x^2 - y^2 = (x - y)(x + y)$$ Here: $$x = (a - 6), y = (b - 8)$$ 7. Step 5: Factorize: $$(a - 6 - (b - 8))(a - 6 + (b - 8)) = (a - 6 - b + 8)(a - 6 + b - 8)$$ Simplify: $$(a - b + 2)(a + b - 14)$$ Final answer: $$(a - b + 2)(a + b - 14)$$