This is essentially Horner's method for calculating the value of a polynomial. [1]
In your example, you calculate the value of the polynomial 8x^5+5x^4+3x^3+6x^2+2x^1+9 with x=10. The value is 853629, of course, but it you perform your calculation modulo 7, you have precisely followed Horner's method in the field F_7. To further simplify the calculation, note (as you did!) that 10 is congruent to 3 mod 7, so use x=3 instead of x=10.
In your example, you calculate the value of the polynomial 8x^5+5x^4+3x^3+6x^2+2x^1+9 with x=10. The value is 853629, of course, but it you perform your calculation modulo 7, you have precisely followed Horner's method in the field F_7. To further simplify the calculation, note (as you did!) that 10 is congruent to 3 mod 7, so use x=3 instead of x=10.
[1] https://en.wikipedia.org/wiki/Horner's_method