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Understanding the concept of the sum of squares is crucial when dealing with polynomial expressions. A sum of squares is an expression of the form \(a^2 + b^2\), where \(a\) and \(b\) are any real numbers or algebraic expressions.
Typically, in mathematics, a sum of squares cannot be factored into simpler polynomial terms with real coefficients. Unlike its counterpart, the difference of squares, a sum of squares doesn't diminish into a product of linear terms in the real number set.
For instance, in the expression \(25s^4 + 16t^2\), each term is a perfect square: \(25s^4 = (5s^2)^2\) and \(16t^2 = (4t)^2\). However, since both components add rather than subtract, we classify this as unfactorable over the reals.

The difference of squares is a well-known and easily recognized format in polynomial expressions. It appears in the form of \(a^2 - b^2\), where both \(a\) and \(b\) are real numbers or variables.
Unlike the sum of squares, a difference of squares can always be factored into \((a + b)(a - b)\). This relation arises because of the middle term cancellation that occurs during expansion.
If we were to have variables substituting the values in our original expression to form \(25s^4 - 16t^2\), it could be rewritten as \((5s^2 - 4t)(5s^2 + 4t)\). This easy factorability highlights the key difference between sum and difference of squares, showing why recognition of the expression's type at the outset is essential.

Factorization over the real numbers involves expressing a polynomial as a product of linear or irreducible polynomial terms with real coefficients. However, not all forms of expressions are conducive to such factorization.
In the case of the expression \(25s^4 + 16t^2\), being a sum of squares, it cannot be broken down into simpler terms within the realm of real coefficients. Real number factorization is thus limited by the nature of the polynomial.
When dealing with real numbers, different techniques such as GCF (Greatest Common Factor), grouping, or applying special products are used to factor expressions. Yet, these techniques fall short with sums of squares without introducing complex numbers or other specialized mathematical fields.