Given $x - y + 1 = 0$, then $y=x+1$ or $x=y-1$. If $1 < y < 2$, then $0 < x < 1$. $E = \sqrt{4x^2 +4y-3} + 2\sqrt{y^2 - 6x - 2y +10}$, $E = \sqrt{4x^2 +4x+4-3} + 2\sqrt{y^2 - 6y+6 - 2y +10}$, $E = \sqrt{4x^2 +4x+1} + 2\sqrt{y^2 - 8y+16}$, $E = 2x+1 + 2(4-y)=2x+1+2(4-x-1)=7$.