I found an elementary solution of this problem, I don't use the fact that a – b is a divisor of P(a) – P(b) when a, b are integers and P is a polynomial with integer coefficients.
Denote by j and m the ages of John and Maria. Conditions are:

f(m) = 0
f(7) = 77
f(j) = 85
m > j > 7

First, if p(a)=b then polynomial p is of the form p(x) = (a-x)q(x)+b, q is polynomial too. Special case, when r is root of polynomial p, then p(x)=(m-x)q(x).

Now, m is root of the polynomial f, and we have:

f(x) = (m-x)u(x) , u(x) is integer coefficient polynomial. Let x=7,

f(7) = (m-7)u(7)
77 = (m-7)u(7)

(m-7) | 77 (*)

Let x=j,

f(j) = (m - j)u(j)
85 = (m - j)u(j)

(m - j) | 85 (**)

But, we know that f(7) = 77. We have another form of f:

f(x) = (x - 7)v(x) + 77 , v(x) is integer coefficient polynomial.

Let x=j,

f(j) = (j - 7)v(j) +77
85 = (j - 7)v(j) + 77
8 = (j - 7)v(j)

(j - 7) | 8 (***)

We conclude:
Because of (*), m € {14,18,84}
Because of (***), j € {8,9,11,15}

Easy to see that is m-j € {3,5,6,7,9,10,69,73,74,76}. Number 5 is only one divisor of 85 (condition **), m = 14 , j = 9

The polynomial must have been of the form (x-7)(x-9)(x-14)Q(x) - 3x^2 + 52x -140.

Thanks for your time.

From f(7) I got 7 - m = -7, -11, or -77 so that m = 14, 18, or 84

From f(j) = 85, I got (j - m) = -5, -17, or -85

Your (j - 7)|8 precludes some combinations. I missed that.

Thank you for the problem