Need a hint on calculating this $n$-th order determinant
Denote the determinant by $d(a_0,\ldots,a_n,x)$. Expand along the first row: $$d(a_0,\ldots,a_n,x)=a_0x^n+d(a_1,\ldots,a_n,x).$$ So by induction:
$$d(a_0,\ldots,a_n,x)=a_0x^n+a_1x^{n-1}+\ldots+a_n.$$
By the way, the matrix is of size $n+1$, not $n$.