Find the definite integral of $f(x)=x(3 x+2)$ over the interval $[0,2]$.

Step 1 ：The definite integral of a function over an interval [a, b] is the area under the curve of the function from a to b. To find this, we need to first find the antiderivative of the function, then evaluate it at b and a, and subtract the two results.

Step 2 ：The antiderivative of \(f(x)=x(3 x+2)\) can be found by applying the power rule for integration, which states that the integral of \(x^n\) is \((1/(n+1))*x^{n+1}\), and the constant multiple rule, which states that the integral of a constant times a function is the constant times the integral of the function.

Step 3 ：Let's find the antiderivative F of f. We have \(F = x^3 + x^2\).

Step 4 ：Now, we evaluate F at 2 and 0. \(F(2) = 2^3 + 2^2 = 12\) and \(F(0) = 0^3 + 0^2 = 0\).

Step 5 ：Subtracting these two results, we find that the definite integral of \(f(x)=x(3 x+2)\) over the interval [0,2] is \(12 - 0 = 12\).

Step 6 ：Final Answer: The definite integral of \(f(x)=x(3 x+2)\) over the interval [0,2] is \(\boxed{12}\).