Given that \(x = 4\sqrt{12} + 5\sqrt{27} - 3\sqrt{75} + \sqrt{300}\)  and \(y = (2 + \sqrt{3})(2 - \sqrt{3}). \text{ If } \frac{x}{y} = a + b\sqrt{3},\) then what is the value of $(a+2b)$ ?

Given:

\(x = 4\sqrt{12} + 5\sqrt{27} - 3\sqrt{75} + \sqrt{300}\)

\(y = (2 + \sqrt{3})(2 - \sqrt{3})\)

\(\frac{x}{y} = a + b\sqrt{3}\)

Formula used:

\(\sqrt{ab} = \sqrt{a}\sqrt{b}\)

\((a + b)(a - b) = a^2 - b^2\)

Calculation:

First, simplify the expression for x:

\(x = 4\sqrt{12} + 5\sqrt{27} - 3\sqrt{75} + \sqrt{300}\)

⇒ \(x = 4\sqrt{4 \times 3} + 5\sqrt{9 \times 3} - 3\sqrt{25 \times 3} + \sqrt{100 \times 3}\)

⇒ \(x = 4 \times 2\sqrt{3} + 5 \times 3\sqrt{3} - 3 \times 5\sqrt{3} + 10\sqrt{3}\)

⇒ \(x = 8\sqrt{3} + 15\sqrt{3} - 15\sqrt{3} + 10\sqrt{3}\)

⇒ \(x = (8 + 15 - 15 + 10)\sqrt{3}\)

⇒ \(x = 18\sqrt{3}\)

Next, simplify the expression for y:

\(y = (2 + \sqrt{3})(2 - \sqrt{3})\)

This is in the form (a + b)(a - b) = a² - b²

⇒ \(y = 2^2 - (\sqrt{3})^2\)

⇒ \(y = 4 - 3\)

⇒ y = 1

Now, find the value of \(\frac{x}{y}\):

\(\frac{x}{y} = \frac{18\sqrt{3}}{1}\)

⇒ \(\frac{x}{y} = 18\sqrt{3}\)

We are given that \(\frac{x}{y} = a + b\sqrt{3}\).

So, \(18\sqrt{3} = a + b\sqrt{3}\)

By comparing the rational and irrational parts:

a = 0

b = 18

a + 2b = 0 + 2 × 18

⇒ a + 2b = 0 + 36

⇒ a + 2b = 36

∴ The correct answer is option 3.