It takes some trial and error
To find the factored form, you know that the format will look like
(x^n +/- a) (x^n +/- b)
Since the expanded form has x^4, the first part of each expression in parentheses will be x^2 because x^2 * x^2 will give you x^4
Then you have to figure out, how will you get -9 in the expanded form? The "a" and "b" parts of each expression will have to multiply to equal -9.
Factors of -9 will be -3*3 or -1*9 or -9*1
Also, consider the 8x^2 part - as it relates to finding the factors of 9. The parts of the expression that result in 8x^2 in the expanded form ...
a*x^2 + b*x^2 = 8x^2
If you think about this, you know the a and b will = -1 and 9 because you also have to add the expressions with x^2 to get 8x^2, and 3 and 3 would not give you the 8x^2 and neither would nor would -9 and 1.
Final factored form is
(x^2 - 1) (x^2 + 9)