Evaluate. (Be sure to check by differentiating!) \[ \int\left(4 t^{3}+8\right) t^{2} d t \]

Step 1 ：Given the integral \(\int(4 t^{3}+8) t^{2} dt\)

Step 2 ：We can use the power rule for integration, which states that the integral of \(x^n dx\) is \((1/(n+1))x^(n+1)\).

Step 3 ：Applying this rule to each term in the polynomial separately, we get \(\frac{2}{3}t^{6} + \frac{8}{3}t^{3}\).

Step 4 ：To check the result, we differentiate the result and see if we get the original function back.

Step 5 ：The derivative of the integral \(\frac{2}{3}t^{6} + \frac{8}{3}t^{3}\) is \(4t^{5} + 8t^{2}\), which is the original function.

Step 6 ：\(\boxed{\frac{2}{3}t^{6} + \frac{8}{3}t^{3}}\) is the final answer.