<s>[INST] How to find the Square Root of a Polynomial $4x^4 + 4x^3 - 11x^2 -6x + 9$
How do you find the square root of this polynomial? I really don't understand. Please provide an easy-to-understand explanation. Thanks.
 [/INST] If you know that the polynomial is a perfect square, then the square root algorithm works. For example

$$\sqrt{4x^4 + 4x^3 - 11x^2 -6x + 9}$$
\begin{array}{lcccccccccccccc}
&&2x^2 && +x && -3\\
&&---&---&---&---&---\\
  &|& 4x^4 & 4x^3 & -11x^2 & -6x & +9\\
2x^2 && 4x^4\\
&&---&---&---\\
&&& +4x^3 & -11x^2 \\
4x^2+x &|& &+4x^3 &+x^2\\
&&&---&---&---&---\\
&&&& -12x^2 &-6x &+9\\
4x^2+ 2x -3 &|&&&-12x^2 &-6x &+9\\
&&&&---&---&---\\
&&&&&&0\\
\end{array}
\begin{array}{ll}
\text{STEP}\;1. &\text{Compute the square root of the leading term (4x^4) and put it,}\\
& \text{(2x^2), in the two places shown.} \\
\text{STEP}\;2. &\text{Subtract and then append the next two terms ($4x^3-11x^2$).}\\
\text{STEP}\;3. &\text{Double the currently displayed quotient $(2x^2 \to 4x^2)$,} \\
& \text{then add the new term, $x$, so that $x(4x^2 + x)$ will "cancel out" the} \\
& \text{$4x^3$ in $4x^3 -11x^2$.}\\
\text{STEP}\;4. &\text{Subtract and then append the next two terms ($-6x+9$).}\\ 
\text{STEP}\;5. &\text{Double the currently displayed quotient} \\
& \text{ $(2x^2 + x \to 4x^2 + 2x)$, then add a new term, $x$,} \\
& \text{then add a new term, $-3$, so that $-3(4x^2 + 2x - 3)$ will "cancel out"} \\
& \text{ the $-12x^2$ in $-12x^2 -6x + 9$.}
\end{array}
</s>