The given expression is:

$\frac{\sqrt{3} + \sqrt{4}}{\sqrt{3} - \sqrt{4}}$

Let's simplify it step by step.

Step 1: Recognize that $\sqrt{4} = 2$.

So the expression becomes:

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

Step 2: Multiply the numerator and the denominator by the conjugate of the denominator.

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

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

Step 3: Simplify the denominator.

$(\sqrt{3})^2 - (2)^2 = 3 - 4 = -1$

Step 4: Expand the numerator.

$(\sqrt{3} + 2)^2 = (\sqrt{3})^2 + 2 \times \sqrt{3} \times 2 + 2^2 = 3 + 4\sqrt{3} + 4 = 7 + 4\sqrt{3}$

Step 5: Put it all together.

Now the expression becomes:

$\frac{7 + 4\sqrt{3}}{-1} = -(7 + 4\sqrt{3}) = -7 - 4\sqrt{3}$

Thus, the simplified expression is:

$\boxed{-7 - 4\sqrt{3}}$

Would you like further details on any step? Here are 5 related questions you might explore:

1. What is the importance of using the conjugate when simplifying such expressions?
2. How do you expand binomials like $(\sqrt{3} + 2)^2$?
3. Can you generalize this process for similar expressions with square roots?
4. What happens if the denominator does not have a difference of squares?
5. How does rationalizing affect the overall value of a fraction?

Tip: Multiplying by the conjugate is a powerful tool for simplifying expressions with radicals in the denominator!