Theory:

Learn to factorize the expression of the form ax2+bx+c.
Procedure to factorize the expression.
 
Step 1: Determine the coefficient \(a, b\) and \(c\).
 
Step 2: Calculate the product of \(a\) and \(c\).  Product \(= ac\) and sum \(= b\).  Thus the middle coefficient is the sum and extreme product is the product value.
  
Step 3: Express the middle term as sum of two terms such that the sum satisfies the middle term and the product satisfies the extreme product.
 
Step 4:  Now group the expression into two factors by taking the common expression outside.
Example:
1. \(x^2+5x+6\)
 
We have \(a =1\), \(b = 5\) and \(c = 6\).
 
Here the product \(=\) \(a \times c\) \(=\) \((1 \times 6)\) \(= 6\) and sum \(= b = 5\).
 
We need to choose two number such that the sum of two numbers is \(5\) and the product of two numbers is \(6\).
 
\((2+3) = 5\) and \((2 \times 3\) \(= 6\)
 
We can write as follows.
 
\(x^2+5x+6\) \(=\) \(x^2+3x+2x+6\)
 
\(=\)\(x(x+3)+2(x+3)\)
 
\(=\) \((x+3)(x+2)\)
 
 
2. \(2x^2-5x-3\)
 
We have \(a = 2\), \(b = -5\) and \(c = -3\).
 
Here the product \(=\) \(a \times c\) \(=\) \((2 \times -3)\) \(=\) \(-6\) and sum \(= b = -5\).
 
We need to choose two number such that the sum of two numbers is \(-5\) and the product of two numbers is \(-6\).
 
\((-6+1) = -5\) and \((-6\times 1)\)) \(=-6\)
 
We can rewrite as follows.
 
\(2x^2-5x-3\) \(=\) \(2x^2-6x+x-3\)
 
\(= 2x(x-3)+1(x-3)\)
 
\(= (x-3)(2x+1)\)