We can first simplify the absolute values in the equation:

|5x - 7 - 2x^2| = 7 - 5x

|x - 7| = 7 - x

Substitute these values back into the equation:

(7 - 5x) + (7 - x) - 2x^2 + 6x - 14 = 0

Simplify and combine like terms:

14 - 5x - x - 2x^2 + 6x - 14 = 0
-2x^2 + x + 5x + 6x - 14 + 14 - 7 = 0
-2x^2 + 12x - 7 = 0

Now, solve for x using the quadratic formula:

x = (-12 ± √(12^2 - 4(-2)(-7))) / 2(-2)
x = (-12 ± √(144 - 56)) / -4
x = (-12 ± √88) / -4
x = (-12 ± 2√22) / -4
x = 3 ± √22

Therefore, the solutions to the equation are x = 3 + √22 and x = 3 - √22.