Recall the identity $a^2 - b^2 = (a - b)(a + b)$. Apply it to $x^2 - 4^2$ to factor the numerator.

Once the numerator is factored, see if any factor in the numerator matches the denominator. If so, and as long as that factor is not zero, you can cancel it and see what remains.

In Desmos, type the expression (x^2 - 16)/(x - 4) to graph the original function. You should see a curve with a hole at $x = 4$ (because the expression is undefined there).

On separate lines, enter y = x - 4, y = x^2 - 4, y = x^2 + 4, and y = x + 4. Compare each of these graphs to the graph of (x^2 - 16)/(x - 4) and look for the one whose graph lies exactly on the same curve everywhere the original expression is defined (except for the hole at $x = 4$). That matching graph represents the equivalent expression.

The numerator $x^2 - 16$ is a difference of squares because $x^2$ is a square and $16 = 4^2$ is also a square.

Use the pattern $a^2 - b^2 = (a - b)(a + b)$ with $a = x$ and $b = 4$:

Substitute the factored form of the numerator into the original expression:

We are told $x \ne 4$, so the denominator $x - 4$ is not zero, which means we are allowed to cancel this factor.

Since $x - 4$ appears in both the numerator and the denominator and is nonzero (because $x \ne 4$), we can cancel it:

So, for all $x \ne 4$, the expression $\dfrac{x^2 - 16}{x - 4}$ is equivalent to $x + 4$.