The given expression is:

\[
a + \frac{1}{2a + 2} + \frac{2a + 2}{3a + 3}
\]

To simplify, follow these steps:

### Step 1: Factor the denominators
- In the second term, \(2a + 2\) can be factored as \(2(a + 1)\).
- In the third term, \(3a + 3\) can be factored as \(3(a + 1)\).

Now the expression becomes:

\[
a + \frac{1}{2(a + 1)} + \frac{2(a + 1)}{3(a + 1)}
\]

### Step 2: Simplify the third term
Since \( (a + 1) \) appears in both the numerator and denominator of the third term, you can cancel it out:

\[
\frac{2(a + 1)}{3(a + 1)} = \frac{2}{3}
\]

Now the expression is:

\[
a + \frac{1}{2(a + 1)} + \frac{2}{3}
\]

### Step 3: Combine the terms
To simplify further, we need to combine the terms. The first term is \(a\), and the second and third terms have fractions.

So, the final simplified expression is:

\[
a + \frac{1}{2(a + 1)} + \frac{2}{3}
\]

This is the simplified form, combining all the terms as far as possible.

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Would you like further steps to solve or expand on this, or do you have any specific questions?

Here are 5 related questions:
1. How do you handle complex rational expressions?
2. What are the rules for factoring binomials?
3. Can you explain how to combine terms with different denominators?
4. How would this expression change if we used different variables?
5. What happens if the denominator is zero in such expressions?

**Tip**: Always check if any terms can be factored or simplified before attempting to combine fractions!

This guide explains how to simplify the rational expression a + 1/(2a+2) + (2a+2)/(3a+3). Follow step-by-step instructions to factor denominators, cancel common terms, and combine the simplified expression. Ideal for students in Grades 9-10 learning about algebra and rational expressions.

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