SOLUTION: Rewrite x^4+2x^3+6x^2+5x+6 in the form (x^2+x)^2+a(x^2+x)+b. Fully factorise your answer. Thanks!

If  is identical to (x^2+x)^2+a(x^2+x)+b,
then any value substituted for x in both will always give the 
same answer.

Substituting x=0 in the first gives



Substituting x=0 in the second gives



So b=6

Substituting x=1 in the first gives



Substituting x=1 in the second gives



So 4+2a+b = 20, and since b=6
   4+2a+6 = 20
    2a+10 = 20
       2a = 10
        a = 5

So

 is identical to  

and we substitute a=5 and b=6:

.

And that's the form desired.

To factor it let u = x²+x



That factors as



Substitute x²+x for u



Edwin