algebra

Here is a start:

(r-2)(s-2).= rs -2r-2s+4

               =  rs -2 (r+s) +4    Are you familiar with Vieta's formulas for quadratics???

Vieta's for quadratics   :   r + s = -b/a   =19    

                                  and   rs = c/a     = 9

  rs -2 (r+s) +4 =

  9 -2(19)+4 = -25

For this, we don't ACTUALLY have to solve for r and s. 

Let us first expand (r-2)(s-2):

$r\cdot{s}+r\cdot{-2}+{-2}\cdot{s}+{-2}\cdot{-2}=rs-2r-2s+4$

if r and s are the roots of y^2 - 19y + 9, then:

$y^2 - 19y + 9 = (y-r)(y-s) = y^2 - (r + s)y + rs$

so r+s=19 and rs=9

going back to the original equation, we have 

$rs-2r-2s+4=9-2(r+s)+4=9-38+4=\boxed{-25}$