Given polynomial p(x) = x^{2010} + αx^{2011} + 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