<s>[INST] Algebraic Expression making perfect square For what value of $a$ and $b$ would $$x^4+2x^3+ax^2+bx+9$$ 
Would give perfect square?
I have found $a=7$ and $b=6$.
How many more are there?
Can we find all of it ?
 [/INST] The given polynomial is a perfect square if and only if 
$$x^4+2x^3+ax^2+bx+9=(x^2+Ax+B)^2$$
for some $A,B$. That is, after expanding and comparing the coefficients, 
$$2x^3+ax^2+bx+9=2Ax^3+(2B+A^2)x^2+2ABx+B^2
\Leftrightarrow
\begin{cases}
A=1\\
2B+A^2=a\\
2AB=b\\
B^2=9
\end{cases}$$
Now it should be easy to find ALL the values of $a$ and $b$ that give you a perfect square.
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