We are given the expression:

$\frac{2\sqrt{3} + a}{\sqrt{3} - 1}$

where $a$ and $b$ are integers, and we are tasked with simplifying the expression.

Step 1: Rationalize the denominator

To simplify this expression, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{3} - 1$ is $\sqrt{3} + 1$.

We multiply as follows:

$\frac{2\sqrt{3} + a}{\sqrt{3} - 1} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1}$

Step 2: Simplify the denominator

The denominator becomes a difference of squares:

$(\sqrt{3} - 1)(\sqrt{3} + 1) = (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2$

Step 3: Expand the numerator

Now, multiply the numerator:

$(2\sqrt{3} + a)(\sqrt{3} + 1) = 2\sqrt{3} \cdot \sqrt{3} + 2\sqrt{3} \cdot 1 + a \cdot \sqrt{3} + a \cdot 1$

This simplifies to:

$= 2(3) + 2\sqrt{3} + a\sqrt{3} + a = 6 + (2 + a)\sqrt{3} + a$

Step 4: Combine the terms

The expression becomes:

$\frac{6 + (2 + a)\sqrt{3} + a}{2}$

Step 5: Split the expression

We can now split the expression as follows:

$= \frac{6}{2} + \frac{(2 + a)\sqrt{3}}{2} + \frac{a}{2}$

This simplifies to:

$= 3 + \frac{(2 + a)\sqrt{3}}{2} + \frac{a}{2}$

Since $a$ is an integer, $\frac{a}{2}$ will only be simplified if $a$ is an even number.

Thus, the simplified expression is:

$3 + \frac{(2 + a)\sqrt{3}}{2} + \frac{a}{2}$

Let me know if you'd like further details or have any questions.

Here are 5 related questions:

1. How do you simplify expressions with irrational denominators in general?
2. Can you simplify the expression if $a = 2$? What about $a = 4$?
3. What is the significance of using the conjugate when simplifying fractions with square roots?
4. How would this expression change if the denominator were $\sqrt{3} + 1$?
5. Can you apply the same method for complex numbers in the denominator?

Tip: Multiplying by the conjugate is a useful trick to eliminate square roots in the denominator, making expressions easier to simplify.