If the polynomial f=x^4+x^2+1 is divided by g=x^2+2x+3 what is the reminder of division ?

To find the remainder if f(x) = x^4+x^2+1 is divided by
g(x) = x^2+2x+1.

 We divide x^4+x^2+1 by
x^2+2x+3.

x^2+2x+3) x^4+0*x^3 +x^2+0*x+1(
x^2-2x+2

                   x^4  +2x^3  
+3x^2

              
-------------------------------

        x^2+2x+3)-2x^3  
-2x^2 + 0*x+1(-2x

                          -2x^3 - 4x^2
-6x

                     
--------------------------------

                     x^2+2x+3)2x^2+6x+1
(2

                                      
2x^2+4x+6

                                     -------------------

                                               
2x -5 is the remainder.

Therefore f(x)/g(x) = (x^4+x^2+1)/
(x^2+2x+3) = x^2-2x+2 is the quotient and  2x-5 is the
remainder.