You can use Manacher's algorithm to get the maximum palindromic substring centered at each index

After using Manacher's for each center use a line sweep from the center to the left and from the center to the right to find for each index the farthest center to it with distance ≤ palin[center]

After that, find the maximum palindrome size for each prefix in the string and for each suffix and the answer would be max(prefix[i] * suffix[i + 1])