1) (x+1)(x+2) = (x-3)(x+4)
Expanding both sides:
x^2 + 3x + 2 = x^2 + 4x - 3x - 12
x^2 + 3x + 2 = x^2 + x - 12
3x + 2 = x - 12
3x - x = -12 - 2
2x = -14
x = -7

2) (3x-1)(2x+7) - (x+1)(6x-5) = 16
Expanding both sides:
6x^2 + 21x - 2x - 7 - (6x^2 - 5x + 6x - 5) = 16
6x^2 + 19x - 7 - 6x^2 +x - 6x + 5 = 16
19x -7 - x - 6x + 5 = 16
12x -2 = 16
12x = 18
x = 1.5

3) 24 - (3y+1)(4y-5) = (11 - 6y)(2y - 1) + 6
Expanding both sides:
24 - 12y+3 - 5 - 6y + 15 = 22y - 11 + 6
24 - 12y - 2y + 5 = 22y - 5
27 - 14y = 22y - 5
27 + 5 = 22y + 14y
32 = 36y
y = 32/36
y = 8/9

4) (6y+2)(5-y) = 47 - (2y-3)(3y-1)
Expanding both sides:
30y -6y^2 + 10 - 2y = 47 - 6y -2y + 3 +6y^2 -2y
30y -6y^2 + 10 - 2y = 47 - 6y -2y + 3 +6y^2 -2y
30y - 6y^2 + 10 - 2y = 47 - 6y^2 + 4y + 3
30y - 6y^2 + 10 - 2y = - 6y^2 + 4y + 50
30y - 2y + 10 = 4y + 50
28y + 10 = 4y + 50
28y - 4y = 50 - 10
24y = 40
y = 40/24
y = 5/3