We are asked to simplify the following expression:

$\frac{4 + \sqrt{7}}{2 - \sqrt{7}}$

To simplify, we need to rationalize the denominator. To do this, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $2 - \sqrt{7}$ is $2 + \sqrt{7}$.

Step 1: Multiply the numerator and denominator by $2 + \sqrt{7}$:

$\frac{4 + \sqrt{7}}{2 - \sqrt{7}} \cdot \frac{2 + \sqrt{7}}{2 + \sqrt{7}} = \frac{(4 + \sqrt{7})(2 + \sqrt{7})}{(2 - \sqrt{7})(2 + \sqrt{7})}$

Step 2: Simplify the denominator using the difference of squares formula:

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

So the denominator simplifies to $-3$.

Step 3: Expand the numerator:

$(4 + \sqrt{7})(2 + \sqrt{7}) = 4(2) + 4(\sqrt{7}) + \sqrt{7}(2) + \sqrt{7}(\sqrt{7})$ $= 8 + 4\sqrt{7} + 2\sqrt{7} + 7$ $= 15 + 6\sqrt{7}$

Step 4: Combine the numerator and denominator:

$\frac{15 + 6\sqrt{7}}{-3}$

Step 5: Simplify by dividing both terms in the numerator by $-3$:

$\frac{15}{-3} + \frac{6\sqrt{7}}{-3} = -5 - 2\sqrt{7}$

Thus, the expression in simplest radical form with a rational denominator is:

$-5 - 2\sqrt{7}$

Would you like a detailed explanation of any step or have any questions?

Here are five related questions for practice:

1. Rationalize the denominator: $\frac{5 + \sqrt{2}}{3 - \sqrt{2}}$
2. Simplify: $\frac{7 + \sqrt{3}}{2 + \sqrt{3}}$
3. Express in simplest form: $\frac{6 - \sqrt{5}}{2 + \sqrt{5}}$
4. Rationalize and simplify: $\frac{1 + \sqrt{6}}{4 - \sqrt{6}}$
5. Simplify: $\frac{3 + \sqrt{8}}{5 - \sqrt{8}}$

Tip: When rationalizing the denominator, always multiply by the conjugate to eliminate square roots from the denominator.