1. **Stating the problem:** We want to understand how to factor algebraic expressions, which means rewriting them as products of simpler expressions. 2. **Common factoring:** This involves finding the greatest common factor (GCF) of all terms and factoring it out. 3. **Factoring simple trinomials:** For expressions like $ax^2 + bx + c$, we look for two numbers that multiply to $ac$ and add to $b$. 4. **Difference of squares:** Expressions like $a^2 - b^2$ factor as $(a - b)(a + b)$. 5. **Perfect square trinomials:** Expressions like $a^2 \, \pm \, 2ab \, + \, b^2$ factor as $(a \pm b)^2$. 6. **Factoring by decomposition:** Break the middle term into two terms whose coefficients multiply to $ac$ and add to $b$, then factor by grouping. 7. **Step-by-step example:** Factor $x^2 - 9x + 14$. 1. Identify $a=1$, $b=-9$, $c=14$. 2. Find two numbers that multiply to $1 \times 14 = 14$ and add to $-9$: these are $-7$ and $-2$. 3. Rewrite the middle term: $x^2 - 7x - 2x + 14$. 4. Factor by grouping: $$x^2 - 7x - 2x + 14 = (x^2 - 7x) + (-2x + 14)$$ $$= x(x - 7) - 2(x - 7)$$ 5. Factor out the common binomial: $$= (x - 7)(x - 2)$$ 8. **Summary:** Factoring rewrites expressions as products, making them easier to solve or simplify. This explanation covers the main factoring techniques and a clear example.