1. Упростите выражение: a) (4a - b)(a - 6b) + a(25b - 3a); б) (2x + 3y)(x - y) - x(x + y); в) 3a(a + 1) + (a + 2)(a - 3); г) 2c(5c - 3) - (c - 2)(c - 4); д) (3a + b)(a - 2b) + (2a + b)(a - 5b); е) (x + 1)(x + 7) - (x + 2)(x + 3); ж) (a - 4)(a + 6) + (a - 10)(a - 2); з) (y - 3)(5 - y) - (4 - y)(y + 6).

1. Упрощение выражений:

1. а)

   \[ (4a - b)(a - 6b) + a(25b - 3a) \]

   \[ = 4a^2 - 24ab - ab + 6b^2 + 25ab - 3a^2 \]

   \[ = (4a^2 - 3a^2) + (-24ab - ab + 25ab) + 6b^2 \]

   \[ = a^2 + 0ab + 6b^2 \]

   \[ = a^2 + 6b^2 \]

2. б)

   \[ (2x + 3y)(x - y) - x(x + y) \]

   \[ = (2x^2 - 2xy + 3xy - 3y^2) - (x^2 + xy) \]

   \[ = 2x^2 + xy - 3y^2 - x^2 - xy \]

   \[ = (2x^2 - x^2) + (xy - xy) - 3y^2 \]

   \[ = x^2 - 3y^2 \]

3. в)

   \[ 3a(a + 1) + (a + 2)(a - 3) \]

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

   \[ = 3a^2 + 3a + a^2 - a - 6 \]

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

   \[ = 4a^2 + 2a - 6 \]

4. г)

   \[ 2c(5c - 3) - (c - 2)(c - 4) \]

   \[ = (10c^2 - 6c) - (c^2 - 4c - 2c + 8) \]

   \[ = 10c^2 - 6c - (c^2 - 6c + 8) \]

   \[ = 10c^2 - 6c - c^2 + 6c - 8 \]

   \[ = (10c^2 - c^2) + (-6c + 6c) - 8 \]

   \[ = 9c^2 - 8 \]

5. д)

   \[ (3a + b)(a - 2b) + (2a + b)(a - 5b) \]

   \[ = (3a^2 - 6ab + ab - 2b^2) + (2a^2 - 10ab + ab - 5b^2) \]

   \[ = (3a^2 - 5ab - 2b^2) + (2a^2 - 9ab - 5b^2) \]

   \[ = 3a^2 - 5ab - 2b^2 + 2a^2 - 9ab - 5b^2 \]

   \[ = (3a^2 + 2a^2) + (-5ab - 9ab) + (-2b^2 - 5b^2) \]

   \[ = 5a^2 - 14ab - 7b^2 \]

6. е)

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

   \[ = (x^2 + 7x + x + 7) - (x^2 + 3x + 2x + 6) \]

   \[ = (x^2 + 8x + 7) - (x^2 + 5x + 6) \]

   \[ = x^2 + 8x + 7 - x^2 - 5x - 6 \]

   \[ = (x^2 - x^2) + (8x - 5x) + (7 - 6) \]

   \[ = 3x + 1 \]

7. ж)

   \[ (a - 4)(a + 6) + (a - 10)(a - 2) \]

   \[ = (a^2 + 6a - 4a - 24) + (a^2 - 2a - 10a + 20) \]

   \[ = (a^2 + 2a - 24) + (a^2 - 12a + 20) \]

   \[ = a^2 + 2a - 24 + a^2 - 12a + 20 \]

   \[ = (a^2 + a^2) + (2a - 12a) + (-24 + 20) \]

   \[ = 2a^2 - 10a - 4 \]

8. з)

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

   \[ = (5y - y^2 - 15 + 3y) - (4y + 24 - y^2 - 6y) \]

   \[ = (-y^2 + 8y - 15) - (-y^2 - 2y + 24) \]

   \[ = -y^2 + 8y - 15 + y^2 + 2y - 24 \]

   \[ = (-y^2 + y^2) + (8y + 2y) + (-15 - 24) \]

   \[ = 10y - 39 \]