How do we factor a polynomial equation?

Hint: Factoring polynomials can be done by splitting the middle term, but when there is a polynomial equation where the highest power or the degree is more than $5$, then the process of factoring becomes more difficult, for polynomials which have a degree lesser than $5$, synthetic division can be used to split the polynomial into polynomials of a lesser degree.

Complete step-by-step answer:
To factor a polynomial is to write a number as a product of linear terms. It is like writing that the number $10$ is the product of $5$ and $2$.
Let’s consider a polynomial equation of degree $2$.
$ \Rightarrow {x^2} + 6x + 9$
Now at first there is no possible way to write this expression in the form of $2$ linear terms therefore, we use a trick which is called grouping.
We will rewrite the same expression but split the middle term into $2$ parts which give us the same sum of the middle term, and their product is equal to the last term in the polynomial.
Now the middle term can be split up fulfilling both these properties, and written as:
$ \Rightarrow {x^2} + 3x + 3x + 9$
It is to be noted that, by doing this the value of the polynomial does not get changed, it is only written in a simplified manner.
Now it is to be solved as $2$ pairs,
In the first pair consisting of the first $2$ terms, we can see that the term $x$ is common therefore, we can take it out as:
$ \Rightarrow x(x + 3)$
And in the second pair consisting of the last $2$ terms, we can see that $3$ can be removed out as common, therefore it can be written as:
$ \Rightarrow 3(x + 3)$
Now on writing the pairs together we get:
$ \Rightarrow x(x + 3) + 3(x + 3)$
Now the term $(x + 3)$ is common in both the terms therefore, we can take it out as common and write it as:

$ \Rightarrow (x + 3)(x + 3)$, which is the factored way of writing the polynomial expression.

Note:
Factoring in a polynomial equation is done to get the value or the possible values of $x$ in that polynomial.
It is to be noted that when the equation is completely factored, there will only be linear terms.
Another point is that not all polynomials can be factored out, some might not have real roots.