We Distribute, Yet Things Multiply

x² + 8x + 16, (x + 4)(x + 4), x² + 2(4x) + 4²

Using (a + b)² = a² + 2ab + b²: (x + 4)² = x² + 2(x)(4) + 4² = x² + 8x + 16. This is also equal to (x + 4)(x + 4) and follows the pattern a² + 2ab + b².

ab + 3b - 2a - 6

New product = (a + 3)(b - 2) = ab + 3b - 2a - 6. Using distributive property: multiply each term in first bracket by each term in second bracket. The change from original product ab is +3b - 2a - 6.

(x - 5)² = x² - 10x + 25, (2a + 3)² = 4a² + 12a + 9, (y + 7)(y - 7) = y² - 49

First three are correct using identities: (a-b)² = a²-2ab+b², (a+b)² = a²+2ab+b², (a+b)(a-b) = a²-b². The last one is wrong: (3x-2)² = 9x² - 12x + 4, not 9x² - 4.

374

For multiplying by 11, add consecutive digits: 34 × 11 = 3(3+4)4 = 374. This comes from 34 × 11 = 34 × (10 + 1) = 340 + 34 = 374. The pattern is: write first digit, add digits, write last digit.