Consider 4 cases :

+ n = 0 => n⁴ - n + 2 = 2 (unsatisfied)

+ n = 1 => n⁴ - n + 2 = 2 (unsatisfied)

+ n = 2 => n⁴ - n + 2 = 16 = 4² (satisfied)

+ n > 2 

\(\circledast n^4-n+2< n^4\)

\(\circledast\left(n^4-n+2\right)-\left(n^2-1\right)^2=n^4-n+2-n^4+2n^2-1\)

\(=2n^2-n+1=2\left(n^2-\dfrac{1}{2}n+\dfrac{1}{2}\right)\)

\(=2\left(n^2-2n.\dfrac{1}{4}+\dfrac{1}{16}+\dfrac{7}{16}\right)=2\left(n-\dfrac{1}{4}\right)^2+\dfrac{7}{8}>0\)

\(\Rightarrow\left(n^2-1\right)^2< n^4-n+2\)

Hence, n⁴ - n + 2 is between two consecutive perfect squares : (n² - 1)² and (n²)², so n⁴ - n + 2 is not a perfect square

So, n = 2

Phan Thanh Tinh Dao Trong Luan