Is $\sum_{x=1}^n (3x^2+x+1) = n^3+2n^2+3n$? I wanna check if the following equation involving a sum is true or false? How do I solve this? Please help me.
$$ \sum_{x=1}^n (3x^2+x+1) = n^3+2n^2+3n$$
for all $n \in \{0,1,2,3, \dots\}$.

Not quite. Note that it fails at $n=1$. A closed form expression for the sum is $n^3+2n^2+2n$. 
Remark: Recall that $\sum_1^n k^2=\frac{n(n+1)(2n+1)}{6}$ and $\sum_1^n k=\frac{n(n+1)}{2}$. And of course $\sum_1^n 1=n$. 
Or else you can prove the result directly by induction. 
