Since  f(x) = x^3 + ax^2 + bx - 3 leaves remainder 1 when divided by x-1, it means,

according to the remainder theorem, that  f(1) = 1,   or


    1^3 + a*1^2 + b*1 - 3 = 1,

or

    a + b = 1 - 1 + 3,

    a + b = 3.      (1)



Since  f(x) = x^3 + ax^2 + bx - 3 leaves remainder -9 when divided by x+1, it means,

according to the remainder theorem, that  f(-1) = -9,   or


    (-1)^3 + a*(-1)^2 + b*(-1) - 3 = -9,

or

    a - b = 1 + 3 -9,

    a - b = -5.     (2)



Thus we have two equations (1) and (2) to find two unknowns "a" and "b".

So we add these equations, and we get  


    2a = 3 + (-5) = -2,  hence,  a = (-2)/2 = -1.


Then from equation (1),  b = 3 - a = 3 - (-1) = 4.


ANSWER.  a= -1;  b= 4.