Let's analyze the given expression as $n \to +\infty$:

$\lim_{n \to +\infty} \left( 4^{\sqrt{n^4}} + 2n^3 - 4^{\sqrt{n^4}} + n^3 \right)$

First, simplify the expression $4^{\sqrt{n^4}}$.

Step 1: Simplify $4^{\sqrt{n^4}}$

Recall that:

$\sqrt{n^4} = n^2$

Thus,

$4^{\sqrt{n^4}} = 4^{n^2}$

Step 2: Substitute into the expression

Now substitute $4^{n^2}$ into the original expression:

$\lim_{n \to +\infty} \left( 4^{n^2} + 2n^3 - 4^{n^2} + n^3 \right)$

This simplifies to:

$\lim_{n \to +\infty} \left( 2n^3 + n^3 \right)$

Step 3: Combine like terms

Now, add the terms $2n^3 + n^3$:

$\lim_{n \to +\infty} 3n^3$

Step 4: Evaluate the limit

As $n \to +\infty$, $3n^3 \to +\infty$. Thus, the final result is:

$\boxed{+\infty}$

Would you like more details or have any questions?

Here are some related questions you might find helpful:

1. How do limits behave for different powers of $n$?
2. Why does $4^{n^2}$ cancel out in the expression?
3. What are common forms of infinity in limits?
4. How does $n^3$ compare with higher-order terms like $4^{n^2}$?
5. How would the result change if the powers of $n$ were different?

Tip: When dealing with limits at infinity, focus on the terms with the largest growth rate, as they often dominate the result.