Factoring a trinomial involves breaking it down into simpler binomials. Let's go step by step using an example trinomial:

Example:

Factor the trinomial $x^2 + 5x + 6$.

Step-by-Step Guide:

1. Identify the form
   A standard trinomial is written as $ax^2 + bx + c$, where:

   • $a$ is the coefficient of $x^2$
   • $b$ is the coefficient of $x$
   • $c$ is the constant term
     In our example, $x^2 + 5x + 6$, we have:
   • $a = 1$
   • $b = 5$
   • $c = 6$

2. Find two numbers that multiply to give $ac$ and add to give $b$
   Here, we need two numbers that:

   • Multiply to $a \times c = 1 \times 6 = 6$
   • Add up to $b = 5$

   The numbers that satisfy both conditions are 2 and 3 because:

   • $2 \times 3 = 6$ (they multiply to 6)
   • $2 + 3 = 5$ (they add to 5)

3. Rewrite the middle term
   Rewrite the middle term $5x$ as the sum of the two numbers found in the previous step:
   $x^2 + 5x + 6 = x^2 + 2x + 3x + 6$

4. Factor by grouping
   Group the terms in pairs: $(x^2 + 2x) + (3x + 6)$ Now, factor out the greatest common factor (GCF) from each pair: $x(x + 2) + 3(x + 2)$

5. Factor out the common binomial
   Both groups contain the common factor $(x + 2)$, so factor it out: $(x + 2)(x + 3)$

Final Answer:

The factored form of $x^2 + 5x + 6$ is $(x + 2)(x + 3)$.

Recap of the steps:

1. Identify the form and coefficients.
2. Find two numbers that multiply to $a \cdot c$ and add to $b$.
3. Rewrite the middle term.
4. Factor by grouping.
5. Factor out the common binomial.

---

Would you like further details or have any questions?

Here are 5 related questions to consider:

1. How do you factor trinomials when $a \neq 1$?
2. What if there are no two numbers that satisfy the condition?
3. How can factoring help solve quadratic equations?
4. How do you factor trinomials with negative constants?
5. What are perfect square trinomials and how are they factored?

Tip: Always check your factors by multiplying them back to see if they match the original trinomial.