2. Преобразуйте в многочлен: 1) a) 3x(3x + 7) - (3x + 1)²; 2) a) (y - 2)(y + 3) - (y - 1)²; 3) a) (p + 1)² - (p + 2)²; 4) a) 4(a + 5)² - (4a² + 40a); 6) 4b(3b + 6) - (3b - 5)(3b + 5); 6) (c - 5)(c - 1) - (c - 6)²; 6) (y - 4)² - (4 - y)(4 + y); 6) (4ab - b²) + 2(a - b)².

2. Преобразование в многочлен:

1. 1) а)

   \[ 3x(3x + 7) - (3x + 1)^2 \]

   \[ = (9x^2 + 21x) - ( (3x)^2 + 2 × 3x × 1 + 1^2 ) \]

   \[ = (9x^2 + 21x) - (9x^2 + 6x + 1) \]

   \[ = 9x^2 + 21x - 9x^2 - 6x - 1 \]

   \[ = (9x^2 - 9x^2) + (21x - 6x) - 1 \]

   \[ = 15x - 1 \]

2. 2) а)

   \[ (y - 2)(y + 3) - (y - 1)^2 \]

   \[ = (y^2 + 3y - 2y - 6) - (y^2 - 2y + 1) \]

   \[ = (y^2 + y - 6) - (y^2 - 2y + 1) \]

   \[ = y^2 + y - 6 - y^2 + 2y - 1 \]

   \[ = (y^2 - y^2) + (y + 2y) + (-6 - 1) \]

   \[ = 3y - 7 \]

3. 3) а)

   \[ (p + 1)^2 - (p + 2)^2 \]

   \[ = (p^2 + 2p + 1) - (p^2 + 4p + 4) \]

   \[ = p^2 + 2p + 1 - p^2 - 4p - 4 \]

   \[ = (p^2 - p^2) + (2p - 4p) + (1 - 4) \]

   \[ = -2p - 3 \]

4. 4) а)

   \[ 4(a + 5)^2 - (4a^2 + 40a) \]

   \[ = 4(a^2 + 10a + 25) - 4a^2 - 40a \]

   \[ = (4a^2 + 40a + 100) - 4a^2 - 40a \]

   \[ = 4a^2 + 40a + 100 - 4a^2 - 40a \]

   \[ = (4a^2 - 4a^2) + (40a - 40a) + 100 \]

   \[ = 100 \]

5. 6) (первый)

   \[ 4b(3b + 6) - (3b - 5)(3b + 5) \]

   \[ = (12b^2 + 24b) - ((3b)^2 - 5^2) \]

   \[ = (12b^2 + 24b) - (9b^2 - 25) \]

   \[ = 12b^2 + 24b - 9b^2 + 25 \]

   \[ = (12b^2 - 9b^2) + 24b + 25 \]

   \[ = 3b^2 + 24b + 25 \]

6. 6) (второй)

   \[ (c - 5)(c - 1) - (c - 6)^2 \]

   \[ = (c^2 - c - 5c + 5) - (c^2 - 12c + 36) \]

   \[ = (c^2 - 6c + 5) - (c^2 - 12c + 36) \]

   \[ = c^2 - 6c + 5 - c^2 + 12c - 36 \]

   \[ = (c^2 - c^2) + (-6c + 12c) + (5 - 36) \]

   \[ = 6c - 31 \]

7. 6) (третий)

   \[ (y - 4)^2 - (4 - y)(4 + y) \]

   \[ = (y^2 - 8y + 16) - (16 - y^2) \]

   \[ = y^2 - 8y + 16 - 16 + y^2 \]

   \[ = (y^2 + y^2) - 8y + (16 - 16) \]

   \[ = 2y^2 - 8y \]

8. 6) (четвертый)

   \[ (4ab - b^2) + 2(a - b)^2 \]

   \[ = 4ab - b^2 + 2(a^2 - 2ab + b^2) \]

   \[ = 4ab - b^2 + 2a^2 - 4ab + 2b^2 \]

   \[ = 2a^2 + (-b^2 + 2b^2) + (4ab - 4ab) \]

   \[ = 2a^2 + b^2 \]