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DETAILS SCALC9 4.4.005.
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Find the general indefinite integral. (Use $C$ for the constant of integration.)
\[
\int\left(3 x^{2}+8 x+9\right) d x
\]
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Step 1 ：The question is asking for the indefinite integral of the function \(3x^2 + 8x + 9\). The indefinite integral of a function is the antiderivative of that function. The antiderivative of a function is the function whose derivative is the original function.

Step 2 ：The antiderivative of \(x^n\) is \(\frac{1}{n+1}x^{n+1}\), and the antiderivative of a constant is the constant times \(x\).

Step 3 ：So, to find the antiderivative of \(3x^2 + 8x + 9\), we can apply these rules to each term separately.

Step 4 ：The antiderivative of the function \(3x^2 + 8x + 9\) is \(x^3 + 4x^2 + 9x\).

Step 5 ：However, the general form of an indefinite integral also includes a constant of integration, typically denoted as \(C\). This is because the derivative of a constant is zero, so when taking the antiderivative, we don't know what constant may have been there originally.

Step 6 ：Therefore, the final answer should be \(x^3 + 4x^2 + 9x + C\).

Step 7 ：Final Answer: \(\boxed{x^3 + 4x^2 + 9x + C}\)