Why cant 81-x to the forth power be factored unti (3+x) squared times (3-x) squared

Below.

Step-by-step explanation:

81 - x^4        Treat as the difference of 2 squares:

= (9 - x^2)(9 + x^2)

We can further factor the 9 - x^2   (difference of 2 squares)

= (3 - x)(3 + x)(9 + x^2).

9 + x^2 can't be factored.

Part A:

 Let's assume a polynomial of the second degree.
 A second-degree polynomial in its standard form is written as:
 
 Where,
 a, b, c: are the polynomial coefficients. They are real numbers other than zero.
 This definition can be used for a polynomial of degree n.
 For example, the following polynomial is written in its standard form:
 

 Part B:

 By definition we have to:
 A set has closure under an operation if the performance of that operation on the members of the set always produces a member of the same set
 This property is related to polynomials since, for example, if we want to multiply polynomials, we obtain as a result a member of the same set. That is, another polynomial.
 Let's put the following polynomials:
 
 Multiplying we have:
 
 The result of multiplying two polynomials, is also a polynomial.