1. Let's start by stating the problem: We want to factor a function, which means expressing it as a product of simpler functions or polynomials. 2. The general formula or approach depends on the type of function. For polynomials, common methods include factoring out the greatest common factor (GCF), using special products (difference of squares, perfect square trinomials), or factoring trinomials. 3. Important rules: - Always look for a GCF first. - Recognize patterns like $a^2 - b^2 = (a-b)(a+b)$. - For quadratic trinomials $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$. 4. Example: Factor the quadratic function $f(x) = x^2 + 5x + 6$. 5. Step 1: Identify $a=1$, $b=5$, and $c=6$. 6. Step 2: Find two numbers that multiply to $1 \times 6 = 6$ and add to $5$. These numbers are $2$ and $3$. 7. Step 3: Rewrite the middle term using these numbers: $x^2 + 2x + 3x + 6$. 8. Step 4: Factor by grouping: $(x^2 + 2x) + (3x + 6) = x(x + 2) + 3(x + 2)$. 9. Step 5: Factor out the common binomial: $(x + 2)(x + 3)$. 10. So, the factored form of $f(x) = x^2 + 5x + 6$ is $$f(x) = (x + 2)(x + 3)$$. This process helps simplify expressions and solve equations more easily.