2020 AIME II Problem 11

Let P(x)=x^{2}-3 x-7, and let Q(x) and R(x) be two quadratic polynomials also with the coefficient of x^{2} equal to 1. David computes each of the three sums P+Q, P+R, and Q+R and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If Q(0)=2, then R(0)=\frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.