1) Expanding both sides of the equation:
(x^2 - 10x + 25) + (x^2 + 8x + 16) = 2x^2
2x^2 - 2x^2 + 25 + 16 - 10x + 8x = 0
41 - 2x = 0
41 = 2x
x = 41/2

2) Using quadratic formula:
a = 5, b = -12, c = 7
x = (-(-12) ± √((-12)^2 - 457)) / 2*5
x = (12 ± √(144 - 140)) / 10
x = (12 ± √4) / 10
x = (12 ± 2) / 10
x = (14 or 10) / 10
x = 1.4 or 1

3) Factorizing:
x^2 - 121 = (x + 11)(x - 11) = 0
x = -11 or x = 11

4) Factorizing:
5x^2 + 20x = 0
5x(x + 4) = 0
x = 0 or x = -4

5) Setting each factor to zero:
x - 2 = 0
x = 2

-x - 1 = 0
x = -1

The solutions for each equation are:
1) x = 41/2
2) x = 1.4 or 1
3) x = -11 or x = 11
4) x = 0 or x = -4
5) x = 2 or x = -1