The rational root test leads to two roots of [tex]f(x)[/tex], [tex]f(-1)=f(-2)=0[/tex], which means both [tex]x+1[/tex] and [tex]x+2[/tex] are factors.
[tex]\dfrac{x^4 -3x^3 - 5x^2 + 21x + 22}{(x+1)(x+2)}=x^2-6x+11[/tex]
The discriminant of the remaining quadratic is negative [tex]((-6)^2-4(11)=-8)[/tex] and so it has no real roots. So
[tex]f(x)=(x+1)(x+2)(x^2-6x+11)[/tex]
