To solve the equation \( x^2 - 4 + x(x - 2) = 0 \), let's first simplify and combine like terms.

Starting with the given equation:
\[ x^2 - 4 + x(x - 2) = 0 \]

Distribute \( x \) in the term \( x(x - 2) \):
\[ x^2 - 4 + x^2 - 2x = 0 \]

Combine like terms:
\[ 2x^2 - 2x - 4 = 0 \]

To simplify, divide the entire equation by 2:
\[ x^2 - x - 2 = 0 \]

Now, we need to factor the quadratic equation \( x^2 - x - 2 = 0 \). We look for two numbers that multiply to \(-2\) (the constant term) and add up to \(-1\) (the coefficient of the linear term).

The numbers \(-2\) and \(1\) satisfy these conditions:
\[ (-2) \cdot 1 = -2 \]
\[ (-2) + 1 = -1 \]

So, we can factor the quadratic as:
\[ (x - 2)(x + 1) = 0 \]

To find the solutions, set each factor equal to zero:
\[ x - 2 = 0 \quad \text{or} \quad x + 1 = 0 \]

Solving these equations gives:
\[ x = 2 \quad \text{or} \quad x = -1 \]

Therefore, the solutions to the equation \( x^2 - 4 + x(x - 2) = 0 \) are:
\[ x = 2 \quad \text{and} \quad x = -1 \]