One way is to expand the righthand side, then match its terms to those of the lefthand side.

x^2 +6x -3 = (x+a)^2 +b

x^2 +6x -3 = (x^2 +2ax +a^2) +b

x^2 +6x -3 = x^2 +2ax +a^2 +b

x^2 +6x -3 = x^2 +2ax +(a^2 +b)

So,

x^2 = x^2 ----(i)

6x = 2ax -----(ii)

-3 = (a^2 +b) -----(iii)

From (ii),

6 = 2a

a = 6/2 = 3 ----***

Substitute that into (iii),

-3 = (3)^2 +b

-3 = 9 +b

-3 -9 = b

b = -12 ---***

Check those in the original equation,

x^2 +6x -3 = (x+a)^2 +b

x^2 +6x -3 =? (x+3)^2 +(-12)

x^2 +6x -3 =? (x^2 +6x +9) -12

x^2 +6x -3 =? x^2 +6x +9 -12

x^2 +6x -3 =? x^2 +6x -3

Yes, so, OK

Therefore, a = 3 and b = -12. -------answer.