If  $x^2+px+q=0$is the quadratic equation whose roots are a-2 and b-2 where a and b are the roots of$x^2-3x+1=0$  then

If a and b are the roots of$x^2-3x+1=0$  then a+b=3 , ab=1

And a-2 , b-2 are roots of  $x^2+px+q=0$

$$\begin{gathered} \Rightarrow a-2+b-2=-p \\ \Rightarrow a+b-4=-p \\ \Rightarrow 3-4=-p \\ \Rightarrow p=1 \\ and \left(a-2\right)\left(b-2\right)=q \\ \Rightarrow ab-\left(a+b\right)2+4=q \\ \Rightarrow 1-3\left(2\right)+4=q \\ \Rightarrow q=-1 \end{gathered}$$