1.
Expand and simplify: (4a - b - 2)(3a - 7) - (3a + 4b)².
Correct answer: 3a² - 34a - 27ab + 7b - 16b² + 14. (4a - b - 2)(3a - 7) = 12a² - 34a - 3ab + 7b + 14. (3a + 4b)² = 9a² + 24ab + 16b². Subtract to get 3a² - 34a - 27ab + 7b - 16b² + 14.
2.
Simplify: (3x - 1)³.
Correct answer: 27x³ - 27x² + 9x - 1. (3x - 1)³ = (3x - 1)(3x - 1)(3x - 1). First, (3x - 1)(3x - 1) = 9x² - 6x + 1. Then (9x² - 6x + 1)(3x - 1) = 27x³ - 27x² + 9x - 1.
3.
Expand and simplify: (5m - 3n)².
Correct answer: 25m² - 30mn + 9n². (5m - 3n)² = (5m - 3n)(5m - 3n) = 25m² - 15mn - 15mn + 9n² = 25m² - 30mn + 9n².
4.
Simplify: (5x - 1)².
Correct answer: 25x² - 10x + 1. (5x - 1)² = (5x - 1)(5x - 1) = 25x² - 5x - 5x + 1 = 25x² - 10x + 1.
5.
To multiply (x + 5)(2x - 1)(2x + 1), the easiest first step would be:
Correct answer: (x + 5)(4x² - 1). Use difference of squares: (2x - 1)(2x + 1) = 4x² - 1. So the easiest first step is (x + 5)(4x² - 1).
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