well this is not the actual way to solve such a question but i find it helpful
we shall first asume that the remainder is zero.
let the other factor be (X^2) + aX + b
therefore
x^(4)+4x^(3)+6x^(2)+4x+5 = ((X^2)+1)((X^2) + aX + b)
expanding the L.H.S we get
X^(4) + aX^(3) + (b+1)X^(2) + aX + b
comparing the two equations
x^(4)+4x^(3)+6x^(2)+4x+5
X^(4) + aX^(3) + (b+1)X^(2) + aX + b
it can be seen that b=5 , then a=4
since the two equations are the same if a=4 and b=6 then
x^(4)+4x^(3)+6x^(2)+4x+5 = ((X^2)+1)((X^2) + 4X + 6)
then
x^(4)+4x^(3)+6x^(2)+4x+5 divided by x^(2)+1 = x^(2) +4x +6