How can we reuse a previously computed palindrome to compute a larger palindrome?

If “aba” is a palindrome, is “xabax” a palindrome? Similarly is “xabay” a palindrome?

Complexity based hint:
If we use brute-force and check whether for every start and end position a substring is a palindrome we have O(n^2) start - end pairs and O(n) palindromic checks. Can we reduce the time for palindromic checks to O(1) by reusing some previous computation.