To rationalize the denominator of the expression \( \frac{4}{4 + \sqrt{x}} \), we multiply the numerator and denominator by the conjugate of the denominator, which is \( 4 - \sqrt{x} \).
The expression becomes:
\[ \frac{4}{4 + \sqrt{x}} \cdot \frac{4 - \sqrt{x}}{4 - \sqrt{x}} = \frac{4(4 - \sqrt{x})}{(4 + \sqrt{x})(4 - \sqrt{x})} \]
Next, we simplify the denominator using the difference of squares formula:
\[ (4 + \sqrt{x})(4 - \sqrt{x}) = 4^2 - (\sqrt{x})^2 = 16 - x \]
Now we can rewrite the expression as:
\[ \frac{4(4 - \sqrt{x})}{16 - x} \]
Distributing the numerator gives:
\[ \frac{16 - 4\sqrt{x}}{16 - x} \]
Thus, the simplified form of the expression is:
\[ \frac{16 - 4\sqrt{x}}{16 - x} \]