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**(a)** Write \(\frac{12 - \sqrt{18}}{3\sqrt{2} + \sqrt{6}}\) in the form \(a\sqrt{b} + c\sqrt{d}\).

To simplify \(\frac{12 - \sqrt{18}}{3\sqrt{2} + \sqrt{6}}\), we need to rationalize the denominator. We do this by multiplying the numerator and the denominator by the conjugate of the denominator.

The conjugate of \(3\sqrt{2} + \sqrt{6}\) is \(3\sqrt{2} - \sqrt{6}\).

So, we multiply the numerator and the denominator by \(3\sqrt{2} - \sqrt{6}\):

\[
\frac{12 - \sqrt{18}}{3\sqrt{2} + \sqrt{6}} \times \frac{3\sqrt{2} - \sqrt{6}}{3\sqrt{2} - \sqrt{6}} = \frac{(12 - \sqrt{18})(3\sqrt{2} - \sqrt{6})}{(3\sqrt{2})^2 - (\sqrt{6})^2}
\]

First, let's simplify the denominator:

\[
(3\sqrt{2})^2 - (\sqrt{6})^2 = 9 \cdot 2 - 6 = 18 - 6 = 12
\]

Next, we expand the numerator:

\[
(12 - \sqrt{18})(3\sqrt{2} - \sqrt{6})
\]

Using the distributive property:

\[
= 12 \cdot 3\sqrt{2} - 12 \cdot \sqrt{6} - \sqrt{18} \cdot 3\sqrt{2} + \sqrt{18} \cdot \sqrt{6}
\]

Simplifying each term:

\[
= 36\sqrt{2} - 12\sqrt{6} - 3\sqrt{36} + \sqrt{108}
\]

We know that \(\sqrt{36} = 6\), so:

\[
= 36\sqrt{2} - 12\sqrt{6} - 18 + \sqrt{108}
\]

We also simplify \(\sqrt{108}\):

\[
\sqrt{108} = \sqrt{36 \cdot 3} = 6\sqrt{3}
\]

So the numerator becomes:

\[
= 36\sqrt{2} - 12\sqrt{6} - 18 + 6\sqrt{3}
\]

Putting it all together:

\[
\frac{36\sqrt{2} - 12\sqrt{6} - 18 + 6\sqrt{3}}{12} = 3\sqrt{2} - \sqrt{6} - \frac{3}{2} + \frac{\sqrt{3}}{2}
\]

So, the simplified form is:

\[
\frac{12 - \sqrt{18}}{3\sqrt{2} + \sqrt{6}} = 3\sqrt{2} - \sqrt{6} - \frac{3}{2} + \frac{\sqrt{3}}{2}
\]

**(b)** Hence, or otherwise, solve \((y + 4\sqrt{3})\sqrt{2} = 24 - \sqrt{6}\) giving your answer in simplest form.

First, distribute \(\sqrt{2}\):

\[
y\sqrt{2} + 4\sqrt{6} = 24 - \sqrt{6}
\]

Next, isolate the terms involving \(y\):

\[
y\sqrt{2} = 24 - \sqrt{6} - 4\sqrt{6}
\]

Combine like terms:

\[
y\sqrt{2} = 24 - 5\sqrt{6}
\]

Divide both sides by \(\sqrt{2}\):

\[
y = \frac{24 - 5\sqrt{6}}{\sqrt{2}}
\]

Rationalize the denominator:

\[
y = \frac{(24 - 5\sqrt{6}) \cdot \sqrt{2}}{2} = \frac{24\sqrt{2} - 5\sqrt{12}}{2}
\]

Simplify \(\sqrt{12}\):

\[
\sqrt{12} = 2\sqrt{3}
\]

So,

\[
y = \frac{24\sqrt{2} - 10\sqrt{3}}{2} = 12\sqrt{2} - 5\sqrt{3}
\]

Therefore, the solution is:

\[
y = 12\sqrt{2} - 5\sqrt{3}
\]

Would you like further details or have any questions? Here are some related questions and a tip for further practice:

1. How do you rationalize the denominator of a fraction involving surds?
2. What are the steps to combine like terms involving surds?
3. How do you simplify expressions involving square roots?
4. Can you show an example of solving an equation involving surds?
5. How do you verify the solution to an equation involving surds?
6. What are common mistakes to avoid when working with surds?
7. How can you apply the properties of square roots to simplify expressions?
8. What are some advanced problems involving surds and their solutions?

**Tip:** Always double-check your work by substituting your solution back into the original equation to ensure it holds true.

Learn how to solve surd equations by rationalizing denominators and simplifying expressions. This problem covers advanced high school level topics including surds, rationalizing techniques, and simplification methods.

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