To solve the equation \(x^2 - 16 - (x - 4) = 0\), let's start by simplifying the expression.

First, distribute the negative sign through the parentheses:

\[ x^2 - 16 - x + 4 = 0 \]

Next, combine like terms:

\[ x^2 - x - 12 = 0 \]

Now, we have a quadratic equation in standard form:

\[ x^2 - x - 12 = 0 \]

To solve this quadratic equation, we can factor it. We need to find two numbers that multiply to \(-12\) and add up to \(-1\). These numbers are \(-4\) and \(3\).

So, we can factor the quadratic equation as follows:

\[ (x - 4)(x + 3) = 0 \]

To find the solutions, set each factor equal to zero:

\[ x - 4 = 0 \quad \text{or} \quad x + 3 = 0 \]

Solving these equations gives:

\[ x = 4 \quad \text{or} \quad x = -3 \]

Therefore, the solutions to the equation \(x^2 - 16 - (x - 4) = 0\) are \(x = 4\) and \(x = -3\).