Concept of Quadratic Expression Factorization
Factorizing a quadratic expression involves breaking it down into simpler expressions (products of binomials) that, when multiplied together, give the original quadratic expression. A general quadratic expression is of the form:
$ax^2 + bx + c$
Here, $a$, $b$, and $c$ are constants. The goal of factorization is to express this quadratic expression as a product of two binomials.
Steps to Factorize

Identify coefficients: Note the coefficients $a$, $b$, and $c$ from the quadratic expression.

Find two numbers: These numbers should multiply to $ac$ (product of the coefficient of $x^2$ and the constant term) and add to $b$ (coefficient of $x$).

Rewrite the middle term: Use the two numbers found to split the middle term $bx$ into two terms.

Factor by grouping: Group the terms into pairs and factor out the greatest common factor (GCF) from each pair.

Combine terms: Express the quadratic expression as a product of two binomials.
Example
Consider the quadratic expression:
$6x^2 + 11x + 3$
 Coefficient identification: $a = 6$, $b = 11$, and $c = 3$
 Find two numbers:
We need numbers that multiply to $6 \times 3 = 18$ and add to $11$. The numbers $9$ and $2$ satisfy these conditions.
 Rewrite middle term: Split $11x$ as $9x + 2x$:
$6x^2 + 9x + 2x + 3$
 Factor by grouping: Group the terms:
$(6x^2 + 9x) + (2x + 3)$
Factor out the GCF from each group:
$3x(2x + 3) + 1(2x + 3)$
 Combine terms: Factor out the common binomial:
$(3x + 1)(2x + 3)$
Therefore, the factorized form of $6x^2 + 11x + 3$ is:
$(3x + 1)(2x + 3)$
Important Points to Remember
 Check the product and sum of the pair of numbers carefully to ensure they fit the form $ac$ and $b$.
 Always factor out the GCF first if a common factor exists for all terms in the quadratic expression.
Special Cases
Perfect Square Trinomial:
If the quadratic expression is a perfect square trinomial, like $x^2 + 6x + 9$, it factors to:
$(x + 3)^2$
Difference of Squares:
If the quadratic expression is a difference of squares, like $x^2  9$, it factors to:
$(x + 3)(x  3)$