We are given the functional equation:

$f(x) + f(x-1) = x^2$

and the additional condition $f(20) = 90$. We are tasked with finding $f(90)$.

Step 1: Use the given functional equation

The functional equation can be rewritten as:

$f(x) = x^2 - f(x-1)$

This allows us to express each value of $f(x)$ in terms of the previous function value, which suggests a recursive approach.

Step 2: Apply known values

We are given $f(20) = 90$. Let's start calculating values backward using the recursion:

For $f(19)$:

$f(20) + f(19) = 20^2 = 400$ $90 + f(19) = 400$ $f(19) = 400 - 90 = 310$

For $f(18)$:

$f(19) + f(18) = 19^2 = 361$ $310 + f(18) = 361$ $f(18) = 361 - 310 = 51$

For $f(17)$:

$f(18) + f(17) = 18^2 = 324$ $51 + f(17) = 324$ $f(17) = 324 - 51 = 273$

Step 3: Continue backward

At this point, we notice that continuing backward step by step would become too cumbersome. However, the recurrence relation $f(x) = x^2 - f(x-1)$ offers a key insight into the structure of the function. We could calculate further terms if needed but would consider a direct or alternate pattern-based approach for values like $f(90)$.