The expression given is:

$\sqrt{x + 1} + \frac{x}{2\sqrt{x + 1}}$

We are asked to rationalize and simplify this expression. Let's walk through the steps:

Step 1: Combine the terms

Both terms have $\sqrt{x + 1}$ in them, but they aren't in a single fraction. Let's write both terms with a common denominator so that we can combine them.

The common denominator will be $2\sqrt{x + 1}$, so we rewrite the first term $\sqrt{x + 1}$ as:

$\sqrt{x + 1} = \frac{2\sqrt{x + 1}}{2}$

Thus, the expression becomes:

$\frac{2\sqrt{x + 1}}{2} + \frac{x}{2\sqrt{x + 1}}$

Step 2: Combine the fractions

Now that both terms have the same denominator, we can combine them into a single fraction:

$\frac{2\sqrt{x + 1}}{2} + \frac{x}{2\sqrt{x + 1}} = \frac{2\sqrt{x + 1} + x}{2\sqrt{x + 1}}$

Step 3: Simplify the numerator

The expression becomes:

$\frac{2\sqrt{x + 1} + x}{2\sqrt{x + 1}}$

At this point, this is the simplified version of the expression. Further simplification would require a different approach, but this is the rationalized and simplified form.

Would you like more details or any further clarification?

Here are five related questions for practice:

1. How would you rationalize an expression with multiple square roots in the denominator?
2. What is the significance of rationalizing the denominator in mathematical expressions?
3. Can you simplify the expression $\frac{x + 3}{\sqrt{x - 2}}$?
4. What steps are involved in combining fractions with different denominators?
5. How can you simplify $\frac{\sqrt{x} + 1}{\sqrt{x}}$?

Tip: When simplifying complex expressions, always check if there’s a common denominator before combining terms. This can help simplify the process significantly!