(x + |xl)2 - 2(x - |xl) - 64= 0.

To solve the equation \((x + |x|)^2 - 2(x - |x|) - 64 = 0\), we first need to consider the absolute value function \(|x|\), which behaves differently based on whether \(x\) is non-negative or negative. We will solve the equation for both cases.

Case 1: \(x \geq 0\)

When \(x \geq 0\), we have \(|x| = x\). Therefore, we can substitute into the equation:

\[ (x + |x|)^2 - 2(x - |x|) - 64 = 0 \]

This simplifies to:

\[ (x + x)^2 - 2(x - x) - 64 = 0 \]

Which further simplifies to:

\[ (2x)^2 - 2(0) - 64 = 0 \]

\[ 4x^2 - 64 = 0 \]

Now, solving for \(x\):

\[ 4x^2 = 64 \] \[ x^2 = 16 \] \[ x = 4 \quad (\text{since } x \geq 0) \]

Case 2: \(x < 0\)

When \(x < 0\), we have \(|x| = -x\). Now substituting into the equation gives:

\[ (x + |x|)^2 - 2(x - |x|) - 64 = 0 \]

This simplifies to:

\[ (x - x)^2 - 2(x - (-x)) - 64 = 0 \]

Which further simplifies to:

\[ 0 - 2(x + x) - 64 = 0 \] \[ -4x - 64 = 0 \]

Now, solving for \(x\):

\[ -4x = 64 \] \[ x = -16 \quad (\text{since } x < 0) \]

Summary of Solutions

Thus, the solutions to the equation \( (x + |x|)^2 - 2(x - |x|) - 64 = 0 \) are:

\[ x = 4 \quad \text{and} \quad x = -16 \]