[tex]\bf \textit{difference of squares}
\\ \quad \\
(a-b)(a+b) = a^2-b^2\qquad \qquad
a^2-b^2 = (a-b)(a+b)\\ \quad \\ \quad \\
% difference of cubes
\textit{difference of cubes}
\\ \quad \\
a^3+b^3 = (a+b)(a^2-ab+b^2)\qquad
(a+b)(a^2-ab+b^2)= a^3+b^3 \\ \quad \\
a^3-b^3 = (a-b)(a^2+ab+b^2)\qquad
(a-b)(a^2+ab+b^2)= a^3-b^3\\\\
-------------------------------[/tex]
[tex]\bf x^6-9x^4-81x^2+729\qquad
\begin{cases}
x^6=x^{2\cdot 3}\\
\qquad (x^2)^3\\
729=9\cdot 9\cdot 9\\
\qquad 9^3\\
81=9\cdot 9\\
\qquad 9^2
\end{cases}\\\\
-------------------------------\\\\[/tex]
[tex]\bf (x^6+729)-(9x^4+81x^2)\implies [(x^2)^3+9^3]-(9x^2x^2+9^2x^2)
\\\\\\\
[\underline{(x^2+9)}(x^4-9x^2+9^2)]\ -\ [9x^2\underline{(x^2+9)}]\impliedby
\begin{array}{llll}
notice\ the\\
\textit{\underline{common factor}}
\end{array}
\\\\\\
(x^2+9)\ [(x^4\underline{-9x^2}+9^2)\underline{-9x^2}]\impliedby \textit{now we add \underline{these two}}
\\\\\\[/tex]
[tex]\bf (x^2+9)\ [x^4-18x^2+9^2]\impliedby
\begin{cases}
\sqrt{x^4}=x^2\\
\sqrt{9^2}=9\\
18x^2=2(x^2)(9)\\
thus\ a\\
\textit{perfect square trinomial}
\end{cases}
\\\\\\
(x^2+9)\ [(x^2)^2-18x^2+9^2]
\\\\\\
(x^2+9)(x^2-9)^2\implies (x^2+9)(x^2-3^2)^2
\\\\\\
(x^2+9)\ [(x-3)(x+3)]^2\implies \boxed{(x^2+9)(x-3)^2(x+3)^2}
\\\\\\
\textit{and I guess you could be redundant and use}
\\\\\\
(x^2+9)(x-3)(x-3)(x+3)(x+3)[/tex]
