When x=-1 2x-3=-5; when x=3 2x-3=3. For continuity, ax+b=-5 when x=-1, so b-a=-5, b=a-5.
For continuity, bx^2-x+a=3 when x=3, so 9b-3+a=3, 9b+a=6=9(a-5)+a=10a-45. So 10a=6+45=51, a=5.1, so b=0.1.
f(x)=(5.1x+0.1 | x<-1
.......(2x-3 | -1<x<3
.......(0.1x^2-x+5.1 | x>3