<s>[INST] Let $F(x)=x^4-bx^3-11x^2+4(b+1)x+a$.Find $a,b$ Let $F(x)=x^4-bx^3-11x^2+4(b+1)x+a$. Find $a,b$
It’s given that $F(x)$ is a complete square of a quadratic polynomial and $(x+2) $ is a factor of $F(x)$
My attempt :
I can write $F(x) = (x+2)^2(Ax+B)^2$
Then putting $x=-2$, I got $a=36$
And by putting  $x=0$, I got $4B^2=36$
Then $B=+3$ or $B=-3$.
Then clearly $A=1$.
Now I have $F(x)=(x+2)^2(x\pm 3)^2$
Now when x=-3 I get b= 2/5
And when x=3 I get b= 2.
So $F(x)=(x+2)^2(x- 3)^2$. 
How I can I properly select whether $B=-3$ or $B =3$
 [/INST] You proved $F(x) = (x+2)^2(x\pm 3)^2$, and if you multiply out the terms, the coefficient of $x^2$ in $F$ will be
$$
\left[x^2\right]F(x) = 4 \pm 24 + 9 = 13 \pm 24
$$
but you know this must be $-11$, so the correct sign is $-$ and you end up with $$F(x) = (x+2)^2(x-3)^2.$$
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