Answer: Option D[tex](9x^2+1)(3x+1)(3x-1)[/tex]Step-by-step explanation:We have the following expression[tex]81x^4-1[/tex]We can rewrite the expression in the following way:[tex](9x^2)^2-1^2[/tex]Remember the following property[tex](a+b)(a-b) = a^2 -b^2[/tex]Then in this case [tex]a=(9x^2)[/tex] and [tex]b=1[/tex]So we have that[tex](9x^2)^2-1^2[/tex][tex](9x^2+1)(9x^2-1)[/tex]Now we can rewrite the expression [tex]9x^2[/tex] as follows[tex](3x)^2[/tex]So[tex](9x^2+1)(9x^2-1) =(9x^2+1)((3x)^2-1^2)[/tex]Then in this case [tex]a=(3x)[/tex] and [tex]b=1[/tex]So we have that[tex](9x^2+1)(9x^2-1) =(9x^2+1)((3x)^2-1^2)[/tex][tex](9x^2+1)(9x^2-1) =(9x^2+1)(3x+1)(3x-1)[/tex]finally the factored expression is:[tex](9x^2+1)(3x+1)(3x-1)[/tex]