X^2-6x/x^2-16 — 2x-16/x^2-16

To simplify the expression \(\frac{x^2 - 6x}{x^2 - 16} - \frac{2x - 16}{x^2 - 16}\), we can start by combining the fractions, since they have a common denominator:

\[ \frac{x^2 - 6x - (2x - 16)}{x^2 - 16} \]

Next, simplify the numerator:

\[ x^2 - 6x - 2x + 16 = x^2 - 8x + 16 \]

Now our expression becomes:

\[ \frac{x^2 - 8x + 16}{x^2 - 16} \]

Next, we factor both the numerator and the denominator.

The numerator \(x^2 - 8x + 16\) can be factored as:

\[ (x - 4)^2 \]

The denominator \(x^2 - 16\) is a difference of squares, which can be factored as:

\[ (x - 4)(x + 4) \]

Now we rewrite the expression:

\[ \frac{(x - 4)^2}{(x - 4)(x + 4)} \]

We can cancel one \((x - 4)\) from the numerator and the denominator (as long as \(x \neq 4\)):

\[ \frac{x - 4}{x + 4} \]

Thus, the simplified expression is:

\[ \frac{x - 4}{x + 4} \quad (x \neq 4, \, x \neq -4) \]

This is the final result.