Find the value of α and β , if the division of x2010+αx2011+β by (x−1)  leaves the remainder zero and by (x+1) leaves the remainder 4.

Please tell if my answer is correct or not.

I don't know too, the answer in  paper was showing both alpha, beta equals -2 however for both of us  its coming -2 and 1, Thus I think this is correct. It might be some printing error.

Given polynomial p(x) = x²⁰¹⁰ + αx²⁰¹¹ + β 

Now, as the remainder is zero when p(x) is divided by (x- 1). So put x = 1. 

(1)²⁰¹⁰ + α(1)²⁰¹¹ + β = 0 

1 + α + β = 0

α + β = -1.... (1)

Similarly, put x = -1, In this case the remainder is 4

(-1)²⁰¹⁰ + α(-1)²⁰¹¹ + β = 4

1 -  α + β = 4

β - α = 3... (2) 

Adding (1) and (2) we get β = 1 and α = -2