16x^4-81x^2=x^2(16x^2-81)=x^2(4x-9)(4x+9).
(4x^2+9)(4x^2-9)=16x^4-81, which is not quite 16x^4-81x^2! Check your question again. I suspect that the area of the rectangle was actually 16x^4-81, so that the width of 4x^2+9 is a factor and the length is 4x^2-9.