Divide the x^4 +2x^3 + 3x + 4 by x^2 +x, finding the quotient and the remainder.
Show that x^2 + 2x + 8 is an irreducible polynomial.
(x^4 +2x^3 + 3x + 4) by (x^2 +x) , those brackets are essential
= (x+1)(x^3 + x^2 - x + 4)/( x(x+1))
= (x^3 + x^2 - x + 4)/x
= x^2 + x - 1 + 4/x
x^2 + 2x + 8 is an irreducible polynomial because it cannot be factored over the rationals.
that is, we cannot find two numbers that have a sum of 2 and a product of 8