express as perfect square x⁴+2x³+3x²-2x+1

Let , x⁴ + 2 x³ + 3 x² + 2 x + 1 = ( x² + a x + b )²

 

RHS = x⁴ + a² x² + b² + 2a x³ + (2ab) x + 2b x² 

 

If we compare the expanded RHS and LHS , we get , 

 

2a = 2 , i.e a = 1  

 

2ab = 2 ,  hence if a =1 , b = 1

 

Hence , we get , x⁴ + 2 x³ + 3 x² + 2 x + 1 = ( x² + x +1 )²

 

To get a perfect square of given polynomial , coefficient of x term should be +2 not -2 .

Hence the coefficient of x term is changed .

If we do not change it is not possible to express the given polynomial as perfect square