The question asks for the simplified form of $\frac{4\sqrt{5}}{3 - \sqrt{5}}$.

To simplify this, we can use the method of multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of $3 - \sqrt{5}$ is $3 + \sqrt{5}$.

So, we multiply both the numerator and denominator by $3 + \sqrt{5}$:

$\frac{4\sqrt{5}}{3 - \sqrt{5}} \times \frac{3 + \sqrt{5}}{3 + \sqrt{5}} = \frac{4\sqrt{5}(3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})}$

Now, let's simplify both the numerator and denominator:

1. Numerator: $4\sqrt{5}(3 + \sqrt{5}) = 4\sqrt{5} \cdot 3 + 4\sqrt{5} \cdot \sqrt{5} = 12\sqrt{5} + 4 \cdot 5 = 12\sqrt{5} + 20$

2. Denominator: Using the difference of squares formula: $(3 - \sqrt{5})(3 + \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4$

Now the fraction becomes: $\frac{12\sqrt{5} + 20}{4}$

We can simplify this by dividing both terms in the numerator by 4: $\frac{12\sqrt{5}}{4} + \frac{20}{4} = 3\sqrt{5} + 5$

So, the simplified form of $\frac{4\sqrt{5}}{3 - \sqrt{5}}$ is $3\sqrt{5} + 5$, which corresponds to option e in the choices.

Do you need further details or have any questions?

Here are 5 related questions to consider:

1. What is the importance of multiplying by the conjugate when simplifying expressions with square roots?
2. How would the process change if the denominator were $3 + \sqrt{5}$ instead of $3 - \sqrt{5}$?
3. Can you apply the same method to simplify $\frac{7\sqrt{3}}{2 - \sqrt{3}}$?
4. What is the difference between rationalizing the denominator and simplifying radicals?
5. How can you check if your final simplified form is correct?

Tip: Always remember to simplify both the numerator and denominator separately before combining terms!