Expanding the left side of the equation:

(24 + x)² + (x + 10)² = 34²
(24 + x)(24 + x) + (x + 10)(x + 10) = 1156
576 + 24x + 24x + x² + 100 + x² + 20x = 1156
2x² + 68x + 676 = 1156
2x² + 68x - 480 = 0

Now we can solve for x using the quadratic formula:

x = (-68 ± √(68² - 4(2)(-480))) / 2(2)
x = (-68 ± √(4624 + 3840)) / 4
x = (-68 ± √8464) / 4
x = (-68 ± 92) / 4

Solving for x, we get two possible solutions:

x = (92 - 68) / 4 = 6
x = (-92 - 68) / 4 = -40/4 = -10

Therefore, the solutions to the equation are x = 6 and x = -10.