To solve the integral:

$\int (5x^2 - 2)(x + 3) \, dx$

Step 1: Expand the integrand

First, distribute the terms inside the parentheses:

$(5x^2 - 2)(x + 3) = 5x^2(x + 3) - 2(x + 3)$

Expanding each term:

$= 5x^3 + 15x^2 - 2x - 6$

Step 2: Integrate term by term

Now, integrate each term separately:

$\int (5x^3 + 15x^2 - 2x - 6) \, dx$

1. $\int 5x^3 \, dx = \frac{5x^4}{4}$
2. $\int 15x^2 \, dx = \frac{15x^3}{3} = 5x^3$
3. $\int -2x \, dx = -x^2$
4. $\int -6 \, dx = -6x$

Step 3: Combine the results

So, the integral becomes:

$\frac{5x^4}{4} + 5x^3 - x^2 - 6x + C$

where $C$ is the constant of integration.

Final Answer:

$\int (5x^2 - 2)(x + 3) \, dx = \frac{5x^4}{4} + 5x^3 - x^2 - 6x + C$

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Would you like further details or have any questions about this solution?

Here are some related questions:

1. How would the result change if the powers of $x$ were different?
2. How do you perform integration by substitution, and when is it useful?
3. Can you perform integration by parts on a similar problem?
4. What happens if the constant of integration is ignored in problems?
5. How can you check the result of the integration?

Tip: Always expand products first when integrating polynomials to simplify the process.