Sum of values of x in exponential equation

Let $A = 2^{2x^2}$ and $B = 2^{x + 15}$.

Note that $A^2 = (2^{2x^2})^2 = 2^{4x^2}$, $B^2 = (2^{x+15})^2 = 2^{2x+30}$, and $A \cdot B = 2^{2x^2} \cdot 2^{x+15} = 2^{2x^2 + x + 15}$.

The given equation can be rewritten as: $2^{4x^2} - 2 \cdot 2^{2x^2 + x + 15} + 2^{2x + 30} = 0$ $A^2 - 2AB + B^2 = 0$ $(A - B)^2 = 0 \implies A = B$

Thus, $2^{2x^2} = 2^{x + 15} \implies 2x^2 = x + 15 \implies 2x^2 - x - 15 = 0$.

Solving the quadratic equation: $(2x + 5)(x - 3) = 0 \implies x = 3 \text{ or } x = -\frac{5}{2}$

Sum of all possible values of $x = 3 + \left(-\frac{5}{2}\right) = \frac{1}{2}$.