<s>[INST] How can I use Fundamental Theorem of Symmetric Polynomials to factor polynomials? 
How can I use The fundamental theorem of symmetric polynomials (or its proof) to factor symmetric polynomials?  

The link I've given to the theorem uses elaborate wordings using 'rings', 'isomorphic', etc.  
I completely understand those objects or describings are needed to have a deep understanding, but could anyone try, if it is possible, to explain simply how I could use the theorem to, e.g., factor
$(a^4+b^4)(a^2+b^2)-(a^3+b^3)^2 = a^2b^2(a^2+b^2-2ab)$  
without understanding what rings are? I only wish to be able to practically use it.
 [/INST] If I understand correctly, the fundamental theorem of symmetric polynomials says a symmetric polynomial of $x_1, x_2, \dots, x_n$ can be written as a polynomial of $$\quad e_1 = \sum_i x_i,\quad e_2 = \sum_{i < j}x_ix_j, \quad \dots, \quad e_n = \prod_i x_i.$$
For example, symmetric polynomial $(a^4+b^4)(a^2+b^2)-(a^3+b^3)^2$ can be written as a polynomial of $x = a + b, y = ab$. In this example, inductively one can get $a^n + b^n$ in terms of $x,y$. Note that $$a^2 + b^2 = (a+b)^2-2ab = x^2-2y,$$ $$a^3 + b^3 = (a^2 + b^2)(a+b) - ab(a+b) = (x^2-2y)x - yx = x^3-3xy,$$ $$a^4+b^4 = (a^3+b^3)(a+b)-ab(a^2+b^2) = (x^3-3xy)x-y(x^2-2y)=x^4-4x^2y+2y^2.$$
Therefore we have $$(a^4+b^4)(a^2+b^2)-(a^3+b^3)^2 = (x^2-2y)(x^4-4x^2y+2y^2)-(x^3-3xy)^2 = x^2y^2-4y^3.$$
In general, once a symmetric polynomial is written in terms of elementary symmetric polynomials, it might be easier to observe factors. In this example, $x^2y^2-4y^3 = y^2(x^2-4y)$. 
Warning One might want to substitute the original variables back to see if certain factors can be factorized further. In the example, $x^2-4y$ is irreducible, however $x^2-4y = a^2+b^2-2ab = (a-b)^2$ is reducible.
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