### Theory:

Learn to factorize the expression of the form $a{x}^{2}+\mathit{bx}+c$.
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.

$$2x^2-5x-3$$ $$=$$ $$2x^2-6x+x-3$$

$$= 2x(x-3)+1(x-3)$$

$$= (x-3)(2x+1)$$