What is factoring completely?

Hint: We will first define what factoring means and then we will consider different types of equations as examples and try to factor them completely. We should be able to express the equations in the simplest form possible such that it is containing a product of its factors.

Complete step by step answer:
The process of factoring is used to solve higher degree expressions. Factoring is very useful in solving algebraic and other types of equations. If an equation is written in terms of addition or subtraction, then it cannot be considered as a factored form but if an expression is written in the form of a product of numbers or variables, then it can be considered as a factored form.
Let us consider three equations,
\[2{{a}^{2}}+b+c\] ……(1)
\[\left( {{a}^{2}}-{{b}^{2}} \right)\left( a-c \right)\] ……(2)
\[\left( a-b \right)\left( a+b \right)\left( a-c \right)\] …….(3)
Now we will analyse \[1^{st}\]equation,
In this equation, the terms are written as addition of three variables, so it is clear from it that it is a non-factored form of the equation.
If we analyse the \[2^{nd}\] equation, the expression is written as product of two terms, so it can be considered as factored form but the term \[\left( {{a}^{2}}-{{b}^{2}} \right)\] can be further factored and can be written as \[\left( a-b \right)\left( a+b \right)\], so the expression in \[2^{nd}\] equation is factored but not completely.
If we see the \[3^{rd}\] equation, the expression is written as a product of three terms, so it is a factored form and we can see that the expression cannot be factored more than it so it is called the completely factored form of the expression.

Note: We have to keep in mind that there is a difference between factored form and completely factored form of an equation. If an equation cannot be factored further than it is completely factored otherwise it cannot be completely factored.