The mathematical expression in the image is:

$\frac{3\sqrt{3} + \sqrt{7}}{\sqrt{7} - 2\sqrt{3}}$

This can be simplified using the method of rationalizing the denominator. Here’s how:

Step 1: Multiply numerator and denominator by the conjugate of the denominator.

The conjugate of $\sqrt{7} - 2\sqrt{3}$ is $\sqrt{7} + 2\sqrt{3}$.

So, we multiply both the numerator and the denominator by $\sqrt{7} + 2\sqrt{3}$:

$\frac{(3\sqrt{3} + \sqrt{7})(\sqrt{7} + 2\sqrt{3})}{(\sqrt{7} - 2\sqrt{3})(\sqrt{7} + 2\sqrt{3})}$

Step 2: Simplify the denominator.

The denominator is a difference of squares:

$(\sqrt{7})^2 - (2\sqrt{3})^2 = 7 - 12 = -5$

Thus, the denominator simplifies to $-5$.

Step 3: Expand the numerator.

Now, we expand the numerator:

$(3\sqrt{3} + \sqrt{7})(\sqrt{7} + 2\sqrt{3}) = 3\sqrt{3} \cdot \sqrt{7} + 3\sqrt{3} \cdot 2\sqrt{3} + \sqrt{7} \cdot \sqrt{7} + \sqrt{7} \cdot 2\sqrt{3}$

Expanding each term:

$= 3\sqrt{21} + 6\cdot3 + 7 + 2\sqrt{21}$

$= 3\sqrt{21} + 18 + 7 + 2\sqrt{21}$

$= 5\sqrt{21} + 25$

Step 4: Write the final expression.

Now, combining the results from the numerator and denominator:

$\frac{5\sqrt{21} + 25}{-5} = -\frac{5\sqrt{21}}{5} - \frac{25}{5}$

$= -\sqrt{21} - 5$

Thus, the simplified form of the given expression is:

$-\sqrt{21} - 5$

Would you like further clarification or details on any step?

Here are some related questions you might find useful:

1. What is the process of rationalizing the denominator?
2. How do you handle conjugates in algebraic fractions?
3. Why does the difference of squares formula simplify the denominator?
4. Can all irrational expressions be simplified using the same method?
5. What are other methods for simplifying complex fractions?

Tip: Always check if the denominator is a binomial with radicals to apply the conjugate method easily for simplification.