The problem is to simplify the expression $\frac{4}{5 - \sqrt{3}}$.

1. Problem Description

The problem is to simplify the expression $\frac{4}{5 - \sqrt{3}}$.

2. Solution Steps

To simplify the given expression, we need to rationalize the denominator. This means we want to eliminate the square root from the denominator. We can do this by multiplying the numerator and denominator by the conjugate of the denominator, which is $5 + \sqrt{3}$.

$\frac{4}{5 - \sqrt{3}} = \frac{4}{5 - \sqrt{3}} \cdot \frac{5 + \sqrt{3}}{5 + \sqrt{3}}$

Multiplying the numerator:

$4(5 + \sqrt{3}) = 20 + 4\sqrt{3}$

Multiplying the denominator:

$(5 - \sqrt{3})(5 + \sqrt{3}) = 5^2 - (\sqrt{3})^2 = 25 - 3 = 22$

So, the expression becomes:

$\frac{20 + 4\sqrt{3}}{22}$

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

$\frac{20 + 4\sqrt{3}}{22} = \frac{2(10 + 2\sqrt{3})}{2(11)} = \frac{10 + 2\sqrt{3}}{11}$

3. Final Answer

$\frac{10 + 2\sqrt{3}}{11}$