b) [tex]x_1^2-2x_1+m-3=0 \Rightarrow x_1^2+12=2x_1+15-m \Rightarrow 2x_2-x_1x_2=2x_1+15-m \Rightarrow 2(x_1-x_2)=m-15-x_1x_2=m-15-(m-3)=-12 \Rightarrow x_1-x_2=-6 \Rightarrow (x_1-x_2)^2=36 \Rightarrow (x_1+x_2)^2-4x_1x_2=36 \Rightarrow 4-4(m-3)=36 \Rightarrow 16-4m=36 \Rightarrow m=-5[/tex]