The image contains an integral with the following expression:

$\int \frac{x^6 + 4x^3}{(x^4 + x^3 + 1)^2} \, dx$

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Here are some related questions you may want to explore:

1. How do you simplify integrals involving polynomial expressions in both the numerator and denominator?
2. What is the method of substitution in integration and how does it apply here?
3. Can integration by parts be used for polynomials like this?
4. How does the presence of a quadratic or cubic term in the denominator affect the choice of integration methods?
5. What is the strategy for handling rational functions in integrals?

Tip: When integrating rational functions, sometimes performing polynomial long division first can simplify the expression, especially if the degree of the numerator is higher than that of the denominator.