A quadratic is graphically represented by a parabola.
The -8 values suggest a vertex midway between x=1 and 2, x=1.5.
Let y=ax²+bx+c be the equation.
When x=1 and 2 we have a+b+c=4a+2b+c=-8.
3a+b=0 so b=-3a. Therefore a-3a+c=-8, c=2a-8.
When x=3, -6=9a-9a+2a-8, 2a=2 so a=1, b=-3 and c=-6.
Therefore y=x²-3x-6.
Substituting x=4, 5, 6, 7 gives the remaining terms in the series, so the nth term is n²-3n-6.