### Explanation

Factoring expressions is a fundamental process in algebra that involves breaking down complex expressions into simpler, multiplicative components. This makes it easier to solve equations, simplify expressions, and understand the properties of mathematical functions.

### Basic Concept

In algebra, a polynomial expression can be factored into products of simpler polynomials. For instance, consider the polynomial:

$ax^2 + bx + c$
This quadratic expression can often be factored into two binomials:

$(ax + m)(bx + n)$
Where $m$ and $n$ are numbers that satisfy certain conditions derived from the original coefficients $a$, $b$, and $c$.

### Factoring Techniques

There are several common techniques for factoring expressions, including:

**Factoring Out the Greatest Common Factor (GCF)**: Identifying the largest factor that is present in each term of the polynomial.

$6x^2 + 9x = 3x(2x + 3)$
**Factoring by Grouping**: Combining terms with common factors and then factoring out the GCF from each group.

$ax + ay + bx + by = a(x + y) + b(x + y) =$
$= (a + b)(x + y)$
**Difference of Squares**: Recognizing expressions that fit the form $a^2 - b^2$ and factoring them as $(a - b)(a + b)$.

$x^2 - 9 = (x - 3)(x + 3)$
**Trinomials**: Factoring expressions of the form $ax^2 + bx + c$.

$x^2 + 5x + 6 = (x + 2)(x + 3)$
### Importance of Factoring

**Simplification**: Factoring helps to simplify complex expressions, making them easier to work with.

**Finding Roots**: Factoring is often used to find the roots of equations, as the factored form makes it easier to set each component to zero and solve for the variable.

**Solving Equations**: Once an expression is factored, solving equations becomes a more straightforward process.

### Example

Consider the polynomial:

$x^2 - 5x + 6$
To factor this, we look for two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3. Therefore, we have:

$x^2 - 5x + 6 = (x - 2)(x - 3)$
In conclusion, the purpose of factoring expressions is to breakdown complex polynomial expressions into simpler factors, which can then be used to simplify, solve, and analyze various algebraic problems. Understanding this fundamental concept is crucial for progressing in mathematics.