<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] Adjusting the coefficient of $x^3,$
$$4x^4+4x^3+\cdots+x^2=(2x^2)^2+2\cdot2x^2\cdot x+x^2$$
Adjusting the coefficient of $x^2$ and the constants
$$4x^4+4x^3-11x^2+\cdots+3^2=(2x^2)^2+2\cdot2x^2\cdot x+x^2-2\cdot2x^2\cdot3+\cdots+3^2$$
Finally,
$$4x^4+4x^3-11x^2-6x+3^2=(2x^2)^2+2\cdot2x^2\cdot x+x^2-2\cdot2x^2\cdot3+3^2-2\cdot x\cdot3=(2x^2+x-3)^2$$
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