how to factorise x^4+4x^3-x^2-16x-12using long division method

Let f(x)=  x^4+4x^3-x^2-16x-12

Clearly, (x+1), (x+2) are factors of of f(x). (Since, f(-1)=f(-2)=0)

Hence, (x+1).(x+2)=x^2+3x+2 is a factor of f(x).

 

Now, divide f(x) by x^2+3x+2.

The remainder will be 0 and quotient will be x^2 + x - 6.

Now, x^2 + x - 6 = (x+3)(x-2)

 

Hence, f(x) = (x+1) (x+2) (x+3) (x-2)