To factorize the expression \( a^3 + \frac{1}{a^3} + 2a + \frac{2}{a} \), let's break it down step by step.

### Step 1: Group the terms
The expression can be grouped as follows:
\[
\left(a^3 + \frac{1}{a^3}\right) + \left(2a + \frac{2}{a}\right)
\]

### Step 2: Factorize each group
Notice that each group resembles a form that can be factored using algebraic identities.

- The first group \( a^3 + \frac{1}{a^3} \) can be recognized as a part of the identity for the sum of cubes. Using the identity:

\[
a^3 + \frac{1}{a^3} = \left(a + \frac{1}{a}\right) \left(a^2 - 1 + \frac{1}{a^2}\right)
\]

But this identity simplifies better when combined with the second group.

- The second group \( 2a + \frac{2}{a} \) can be factored as:

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

### Step 3: Combine and factor further
Now, combine the factorizations from both groups:

\[
\left(a + \frac{1}{a}\right) \left(a^2 - 1 + \frac{1}{a^2}\right) + 2\left(a + \frac{1}{a}\right)
\]

We can factor out the common term \( \left(a + \frac{1}{a}\right) \):

\[
\left(a + \frac{1}{a}\right)\left[\left(a^2 - 1 + \frac{1}{a^2}\right) + 2\right]
\]

Simplify the expression inside the brackets:

\[
\left(a + \frac{1}{a}\right)\left(a^2 + \frac{1}{a^2} + 1\right)
\]

### Final Factorized Form:
Thus, the factorized form of the expression is:

\[
\boxed{\left(a + \frac{1}{a}\right)\left(a^2 + \frac{1}{a^2} + 1\right)}
\]

Would you like more details or have any questions?

Here are five related questions you might find helpful:

1. How do you factorize expressions involving the difference of cubes?
2. What are the key algebraic identities used in factorization?
3. How do you simplify complex fractions like \( \frac{1}{a^3} \)?
4. Can you factorize \( a^4 + \frac{1}{a^4} + 2a^2 + \frac{2}{a^2} \) in a similar way?
5. What is the importance of recognizing patterns in algebraic expressions?

**Tip:** Always check for common factors across terms before applying more complex factorization techniques.

Learn how to factorize the expression a^3 + 1/a^3 + 2a + 2/a with a detailed step-by-step solution. This problem involves algebraic identities and advanced factorization techniques. Suitable for advanced high school students and beyond.

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