The sum of the roots of the equation, $$x^2 + |2x - 3| - 4 = 0$$, is:

We are given the equation $$ x^2 + |2x - 3| - 4 = 0 $$. The absolute value $$ |2x - 3| $$ changes behavior at the point where $$ 2x - 3 = 0 $$, which is $$ x = \frac{3}{2} $$. Therefore, we need to solve the equation in two cases: when $$ x < \frac{3}{2} $$ and when $$ x \geq \frac{3}{2} $$.

Case 1: $$ x < \frac{3}{2} $$

For $$ x < \frac{3}{2} $$, $$ 2x - 3 < 0 $$, so $$ |2x - 3| = -(2x - 3) = -2x + 3 $$. Substituting this into the equation:

$$ x^2 + (-2x + 3) - 4 = 0 $$

Simplify:

$$ x^2 - 2x + 3 - 4 = 0 $$

$$ x^2 - 2x - 1 = 0 $$

Solve this quadratic equation using the quadratic formula $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$, where $$ a = 1 $$, $$ b = -2 $$, and $$ c = -1 $$.

Discriminant $$ D = b^2 - 4ac = (-2)^2 - 4(1)(-1) = 4 + 4 = 8 $$.

So, $$ x = \frac{2 \pm \sqrt{8}}{2} = \frac{2 \pm 2\sqrt{2}}{2} = 1 \pm \sqrt{2} $$.

The roots are $$ x = 1 + \sqrt{2} $$ and $$ x = 1 - \sqrt{2} $$.

Since this case is for $$ x < \frac{3}{2} = 1.5 $$, we check:

$$ 1 + \sqrt{2} \approx 1 + 1.414 = 2.414 > 1.5 $$ → not valid.

$$ 1 - \sqrt{2} \approx 1 - 1.414 = -0.414 < 1.5 $$ → valid.

Thus, only $$ x = 1 - \sqrt{2} $$ is valid in this case.

Case 2: $$ x \geq \frac{3}{2} $$

For $$ x \geq \frac{3}{2} $$, $$ 2x - 3 \geq 0 $$, so $$ |2x - 3| = 2x - 3 $$. Substituting:

$$ x^2 + (2x - 3) - 4 = 0 $$

Simplify:

$$ x^2 + 2x - 3 - 4 = 0 $$

$$ x^2 + 2x - 7 = 0 $$

Solve using quadratic formula: $$ a = 1 $$, $$ b = 2 $$, $$ c = -7 $$.

Discriminant $$ D = 2^2 - 4(1)(-7) = 4 + 28 = 32 $$.

So, $$ x = \frac{-2 \pm \sqrt{32}}{2} = \frac{-2 \pm 4\sqrt{2}}{2} = -1 \pm 2\sqrt{2} $$.

The roots are $$ x = -1 + 2\sqrt{2} $$ and $$ x = -1 - 2\sqrt{2} $$.

Check $$ x \geq 1.5 $$:

$$ -1 + 2\sqrt{2} \approx -1 + 2 \times 1.414 = -1 + 2.828 = 1.828 > 1.5 $$ → valid.

$$ -1 - 2\sqrt{2} \approx -1 - 2.828 = -3.828 < 1.5 $$ → not valid.

Thus, only $$ x = -1 + 2\sqrt{2} $$ is valid in this case.

The roots of the equation are $$ x = 1 - \sqrt{2} $$ and $$ x = -1 + 2\sqrt{2} $$. The sum of the roots is:

$$ (1 - \sqrt{2}) + (-1 + 2\sqrt{2}) = 1 - \sqrt{2} - 1 + 2\sqrt{2} = (1 - 1) + (-\sqrt{2} + 2\sqrt{2}) = 0 + \sqrt{2} = \sqrt{2} $$

Comparing with the options:

A. 2

B. -2

C. $$ \sqrt{2} $$

D. $$ -\sqrt{2} $$

Hence, the correct answer is Option C.