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!