Let $F(x)=x^4-bx^3-11x^2+4(b+1)x+a$.Find $a,b$

Question

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 $B =3$

Answer 1

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 $$F(x) = (x+2)^2(x-3)^2.$$