2010/053) If r and s are the roots of x^2 + x + 7 = 0, compute 2r^2 + rs + s^2 + r + 7

2r^2 + rs + s^2 + r + 7 =
r^2+rs + s^2 + (r^2+r+7)
= r^2+rs + s^2 as r^2+r+ 7 = 0 as r is a root
= (r+s)^2-rs ..1

now as x^2+x+7 has root r and s so rs = 7 and r+s = - 1

so from (1) we get given expression = 1-7 = - 6