You have:
y = x^2 +5x-3
and
y-x = 2
replace y in the 2nd equation with what Y equals in the first equation:
x^2 +5x-3 - x = 2
Subtract 2 from both sides of the equation:
x^2 +5x-3 - x -2 = 0
Simplify:
x^2 +4x -5 = 0
Factor:
(x-1) (x+5) = 0
Solve for the x's:
1-1 = 0
-5 +5 = 0
X = 1, -5