To solve any polynomial, without using the calculator, factoring play a very vital step.
In order to factor polynomial expressions, there are different approaches which can be used to simplify the procedure.  While all of these approaches are not used for each problem, it is best to examine the given expression for the possible method needed to be used in each situation.
The first step that has to be checked, is to find if there exists any common term in all the terms present in the polynomial. The common term taken will be the greatest common factor (GCF) of all the terms. The main purpose of factoring out GCF, is
to make the expressions more simple to factorize.
GCF can be defined as the largest term that divides all the polynomial term without a reminder.
For example the GCF of the polynomial x2-5x is x as x is common both the terms.

Difference of squares
There exists some polynomial which contains atleast two square terms. Then it can be factored using special formulas.
The general form is $a^{2}-b^{2}=(a-b)(a+b)$
This type of factoring is also called Factoring with Difference of two squares (DOTS)
For example,
y2-25
Here both terms are square terms, separated by a negative.
So we can use the form $a^ {2}-b^ {2} = (a-b) (a+b) $
$y^ {2}-25= y^ {2}-5^ {2} $
              =(y-5) (y+5)

Quadratic Trinomial
There are polynomial which are both quadratic and has three terms in it. Its general form is $ax^ {2} +bx+c$
Let check how this can be factored when the coefficient of $x^ {2}$ is $1$, that is
$a=1$.
For example
$x^{2}-6x+8$
To factor this we need two numbers, such that its sum is $-6$ and its product is $8$.
To find this first lets see different ways of factoring positive $8$
$8= 1*8$
    = $-1*-8$
    = $2*4$
    = $-2*-4$
In this the only numbers such that its sum is -6 is $-2+(-4)= -6$
So
$x^{2}-6x+8=(x-2)(x-4)$

Perfect square trinomial
There are certain trinomial in the form $a^ (2) +2ab +b^ {2} $.
Then it can be reduced to the form $(a+b) ^ {2} $
For example:
x2+10x+25
Here
x2+10x+25= x2+2(5)x+52
So it can be reduced to
x2+2(5)x+52=(x+5)2
                           =(x+5)(x+5)