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The expression presented in the exercise is a 'Sum of Squares'. To identify a sum of squares, look for terms in the form of \((x^2 + y^2)\). In this instance, \(36a^2b^2 + 169c^2\) is presented. This type of expression is often encountered in algebra. Notably, it does not have real number factors like a difference of squares does.

When dealing with a sum of squares, one common approach is to rewrite each term as a perfect square. You can do this by representing each term individually under square brackets. For example, recognize \(36a^2b^2\) as \((6ab)^2\), and \(169c^2\) as \((13c)^2\).

If the expression was a difference of squares, such as \(x^2 - y^2\), it could be factored into \(x + y)(x - y)\). However, with a sum of squares like this one, special techniques are required to factor it, often involving complex numbers.

Some algebraic expressions, like the sum of squares, are classified as 'Non-factorable Expressions' over the real numbers. This means that within the realm of real numbers, you cannot break down the expression further into simpler polynomial terms. For example, the expression \(36a^2b^2 + 169c^2\) does not have any real number outcomes that will simplify or change it through factorization.

Commonly, students might attempt to factor these by either incorrect methods or by trying to apply difference of squares formulas mistakenly. It's important to understand:

• Sum of squares can't be factored using real numbers.
• There are no real roots or linear factors present without employing imaginary or complex components.

This highlights why knowing when expressions are non-factorable is essential. Knowing that an expression is non-factorable prevents missteps in finding unnecessary "solutions" that aren't relevant or applicable under real arithmetic.

In algebra, encountering non-factorable expressions over the real numbers leads us to consider 'Complex Numbers'. Complex numbers involve an imaginary unit denoted as \(i\), where \(i^2 = -1\). This provides a toolkit to manipulate expressions that can't be tackled through traditional real-number methods alone.

Consider our sum of squares expression, \(36a^2b^2 + 169c^2\). This could theoretically be factored using complex numbers into \((6ab + 13ci)(6ab - 13ci)\), turning the sum into a difference of squares format using complex terms.

• Complex numbers expand usual algebra beyond the limitations of real number arithmetic.
• Using the imaginary unit, \(i\), allows for solutions to equations and manipulations previously impossible in the real number set.

As tempting as it can be, applying complex numbers often ventures beyond typical high school curriculum. Nonetheless, it's a vital component in problem-solving across various advanced fields in mathematics and engineering.