{a -> Subscript[x, i], b -> Subscript[x, i - 1], c -> Subscript[x, i+ 1]}
The above means a is x and b is x-h and c is x+h. Then why not just write
Clear[x, a, b, c];
r = Integrate[(x - a)^2 (x - c)^2, {x, b, c}];
r /. {a -> x, b -> x - h, c -> x + h} // Simplify
which gives
(16 h^5)/15
