Question 1040043
One polynomial satisfying the two conditions is {{{f(x) = 2x^2-(41/5)x-12/5}}}.  In this case the remainder is {{{-(3/5)(2x+9)}}}:

{{{(2x^2-(41/5)x-12/5)/(2x^2-7x+3) = 1-((3/5)(2x+9))/(2x^2-7x+3)}}}.