145. Разложите на множители: 1) x² + 2xy + y² – 49; 2) a² – 9b² + 6bc - c²; 3) х³y² – xy -x^3 + x; 4) a³+8-a²-2a; 5) b⁶ - 4b⁴ +12b² - 9; 6) m³ +27n³ + m² + 6mn+9n²; 7) a² + 2ab + b² - c² + 4cd-4d²; 8) a² – b² + 4a + 4.

Решение:

1. x² + 2xy + y² – 49 = (x + y)² – 7² = (x + y – 7)(x + y + 7)
2. a² – 9b² + 6bc - c² = a² – (9b² - 6bc + c²) = a² – (3b – c)² = (a – 3b + c)(a + 3b – c)
3. x³y² – xy - x³ + x = xy(x²y – 1) - x(x² – 1) = xy(x²y – 1) - x(x – 1)(x + 1) = x(x²y² - y - x² + 1)
4. a³ + 8 – a² - 2a = a³ – a² – 2a + 8 = (a³ + 8) – (a² + 2a) = (a + 2)(a² – 2a + 4) – a(a + 2) = (a + 2)(a² – 2a + 4 – a) = (a + 2)(a² – 3a + 4)
5. b⁶ - 4b⁴ +12b² - 9 = (b⁶ +12b² + 36) - (4b⁴ -45)
6. m³ +27n³ + m² + 6mn+9n² = (m + 3n)(m² - 3mn + 9n²) + (m + 3n)² = (m + 3n)(m² - 3mn + 9n² + m + 3n)
7. a² + 2ab + b² - c² + 4cd-4d² = (a² + 2ab + b²) – (c² - 4cd + 4d²) = (a + b)² – (c – 2d)² = (a + b – c + 2d)(a + b + c – 2d)
8. a² – b² + 4a + 4 = (a² + 4a + 4) – b² = (a + 2)² – b² = (a + 2 – b)(a + 2 + b)