To solve each of these equations for x, we need to factor out x or x² and set each factor equal to 0.

1) 10x² + 5x = 0
Factor out x: x(10x + 5) = 0
Set each factor equal to 0:
x = 0 or 10x + 5 = 0
Solve for x: x = 0 or 10x = -5, x = -5/10, x = -1/2

2) x - x² = 0
Factor out x: x(1 - x) = 0
Set each factor equal to 0:
x = 0 or 1 - x = 0
Solve for x: x = 0 or x = 1

3) 6x² + 24x = 0
Factor out 6x: 6x(x + 4) = 0
Set each factor equal to 0:
6x = 0 or x + 4 = 0
Solve for x: x = 0 or x = -4

4) x² - 25 = 0
Factor as the difference of squares: (x + 5)(x - 5) = 0
Set each factor equal to 0:
x + 5 = 0 or x - 5 = 0
Solve for x: x = -5 or x = 5

Therefore, the solutions for each equation are:
1) x = 0, -1/2
2) x = 0, 1
3) x = 0, -4
4) x = -5, 5