15. Упростите выражение: 1) (b + 6)(b – 6) – 3b(b + 2); 2) (a + 3)(a – 3) – 2a(4 + a); 3) (x – 4)² – 6; 4) (x – 3)² – 8; 5) (2a – 3b)² – 4a(a – 6b); 6) (3m – 7n)² – 9m)(m – 5n).

Решение:

1. \( (b+6)(b-6) - 3b(b+2) = (b^2 - 36) - (3b^2 + 6b) = b^2 - 36 - 3b^2 - 6b = -2b^2 - 6b - 36 \)
2. \( (a+3)(a-3) - 2a(4+a) = (a^2 - 9) - (8a + 2a^2) = a^2 - 9 - 8a - 2a^2 = -a^2 - 8a - 9 \)
3. \( (x-4)^2 - 6 = (x^2 - 8x + 16) - 6 = x^2 - 8x + 10 \)
4. \( (x-3)^2 - 8 = (x^2 - 6x + 9) - 8 = x^2 - 6x + 1 \)
5. \( (2a-3b)^2 - 4a(a-6b) = (4a^2 - 12ab + 9b^2) - (4a^2 - 24ab) = 4a^2 - 12ab + 9b^2 - 4a^2 + 24ab = 12ab + 9b^2 \)
6. \( (3m-7n)^2 - 9m)(m-5n) = (9m^2 - 42mn + 49n^2) - (9m^2 - 45mn) = 9m^2 - 42mn + 49n^2 - 9m^2 + 45mn = 3mn + 49n^2 \)

Ответ: 1) -2b² - 6b - 36; 2) -a² - 8a - 9; 3) x² - 8x + 10; 4) x² - 6x + 1; 5) 12ab + 9b²; 6) 3mn + 49n².