1) Let y = x^2
Therefore, the equation becomes y^2 + y - 20 = 0
(y + 5)(y - 4) = 0
y = -5 or y = 4
Case 1: y = -5
x^2 = -5
This has no real solution.
Case 2: y = 4
x^2 = 4
x = ±2
Therefore, the solutions to the equation x^4 + x^2 - 20 = 0 are x = 2 or x = -2.

2) Let y = x^2
Therefore, the equation becomes y^2 - 5y + 4 = 0
(y - 4)(y - 1) = 0
y = 4 or y = 1
Case 1: y = 4
x^2 = 4
x = ±2
Case 2: y = 1
x^2 = 1
x = ±1
Therefore, the solutions to the equation x^4 - 5x^2 + 4 = 0 are x = 2, x = -2, x = 1, or x = -1.

3) Let y = x^2
The equation becomes y^2 - 4y - 5 = 0
(y - 5)(y + 1) = 0
y = 5 or y = -1
Case 1: y = 5
x^2 = 5
x = ±√5
Case 2: y = -1
x^2 = -1
This has no real solution.
Therefore, the solutions to the equation x^4 - 4x^2 - 5 = 0 are x = √5 or x = -√5.

4) Let y = x^2
The equation becomes y^2 + 3y - 4 = 0
(y + 4)(y - 1) = 0
y = -4 or y = 1
Case 1: y = -4
x^2 = -4
This has no real solution.
Case 2: y = 1
x^2 = 1
x = ±1
Therefore, the solutions to the equation x^4 + 3x^2 - 4 = 0 are x = 1 or x = -1.