1. Let's begin by stating the problem: how to factorize algebraic expressions. 2. Factorization means expressing an algebraic expression as a product of its factors. 3. A common approach is to look for the greatest common factor (GCF) first. For example, in $6x^2 + 9x$, the GCF is $3x$, so we write $$6x^2 + 9x = 3x(2x + 3)$$ 4. Another method is to factor quadratic expressions like $ax^2 + bx + c$. For example, factorize $x^2 + 5x + 6$. 5. We look for two numbers that multiply to $6$ (the constant term) and add to $5$ (the coefficient of $x$). The numbers are $2$ and $3$. 6. So, we write $$x^2 + 5x + 6 = (x + 2)(x + 3)$$ 7. For expressions where the leading coefficient $a \neq 1$, like $2x^2 + 7x + 3$, we use splitting the middle term: - Multiply $a$ and $c$: $2 \times 3 = 6$ - Find two numbers that multiply to $6$ and add to $7$: $6$ and $1$ - Rewrite as $2x^2 + 6x + 1x + 3$ - Factor by grouping: $$2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)$$ 8. Also, recognize special factorization patterns: - Difference of squares: $a^2 - b^2 = (a - b)(a + b)$ - Perfect square trinomials: $a^2 \pm 2ab + b^2 = (a \pm b)^2$ 9. Practice is key! Always start by searching for a GCF and then look for recognizable patterns or use methods like grouping or quadratic factorization. Final answer: Factoring is rewriting expressions as products by extracting common factors, recognizing patterns, or using systematic methods such as splitting terms.