To simplify the expression \( 3(x^2 + 1)^{\frac{2}{3}} + 4x(x^2 + 1)^{-\frac{1}{3}} \), let's look for common factors:

### Step 1: Identify Common Factors
Notice that both terms contain \((x^2 + 1)\) raised to a power. The expression can be factored by taking the lowest power of \((x^2 + 1)\), which is \((x^2 + 1)^{-\frac{1}{3}}\), out of both terms:

\[
3(x^2 + 1)^{\frac{2}{3}} + 4x(x^2 + 1)^{-\frac{1}{3}} = (x^2 + 1)^{-\frac{1}{3}} \left[ 3(x^2 + 1)^{\frac{2}{3} + \frac{1}{3}} + 4x \right]
\]

### Step 2: Simplify the Expression Inside the Brackets
Add the exponents inside the bracket:

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

This simplifies the expression to:

\[
(x^2 + 1)^{-\frac{1}{3}} \left[ 3(x^2 + 1) + 4x \right]
\]

### Step 3: Distribute and Combine Like Terms
Expand the expression inside the brackets:

\[
3(x^2 + 1) + 4x = 3x^2 + 3 + 4x
\]

So the simplified expression is:

\[
(x^2 + 1)^{-\frac{1}{3}} (3x^2 + 4x + 3)
\]

### Final Answer
The simplified expression is:

\[
(x^2 + 1)^{-\frac{1}{3}} (3x^2 + 4x + 3)
\]

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Would you like more details or have any questions? 

Here are 5 related questions you might find interesting:
1. How would you simplify expressions with different bases and exponents?
2. What happens if the exponent inside the power is negative or zero?
3. Can you factor quadratic expressions like \(3x^2 + 4x + 3\) further?
4. What techniques can you use to combine or simplify rational expressions?
5. How do you differentiate or integrate functions involving expressions like \((x^2 + 1)^{-\frac{1}{3}}\)?

**Tip:** Always check for common factors before simplifying an expression; it often makes the process easier and more straightforward!

Learn how to simplify the expression 3(x^2 + 1)^2/3 + 4x(x^2 + 1)^-1/3 with a step-by-step solution. This problem involves exponentiation and rational expressions, suitable for high school students.

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