To solve the equation 2(x^2 - 7) = 2^4 + 20, we can follow these steps:

1. Simplify both sides of the equation:
2(x^2 - 7) = 2^4 + 20
2x^2 - 14 = 16 + 20

2. Combine like terms:
2x^2 - 14 = 36

3. Move all terms to one side of the equation to set it equal to zero:
2x^2 - 14 - 36 = 0
2x^2 - 50 = 0

4. Divide both sides of the equation by 2 to isolate the variable:
(2x^2 - 50) / 2 = 0 / 2
x^2 - 25 = 0

5. Factor the quadratic equation:
(x - 5)(x + 5) = 0

6. Set each factor equal to zero and solve for x:
x - 5 = 0 or x + 5 = 0
x = 5 or x = -5

Therefore, the solutions to the equation 2(x^2 - 7) = 2^4 + 20 are x = 5 and x = -5.