Factor the trinomial:
5, x, squared, plus, 23, x, plus, 12


To factor the trinomial \( 5x^2 + 23x + 12 \), we will use the **trial and error method** (or the **AC method**). Here's the step-by-step process:

### Step 1: Multiply the coefficient of \( x^2 \) (5) by the constant term (12):
\[
5 \times 12 = 60
\]

### Step 2: Find two numbers that multiply to 60 and add up to the middle coefficient (23):
The numbers are 3 and 20 because:
\[
3 \times 20 = 60 \quad \text{and} \quad 3 + 20 = 23
\]

### Step 3: Rewrite the middle term \( 23x \) as \( 3x + 20x \):
\[
5x^2 + 3x + 20x + 12
\]

### Step 4: Group the terms in pairs:
\[
(5x^2 + 3x) + (20x + 12)
\]

### Step 5: Factor each group:
\[
x(5x + 3) + 4(5x + 3)
\]

### Step 6: Factor out the common binomial \( (5x + 3) \):
\[
(5x + 3)(x + 4)
\]

Thus, the factored form of \( 5x^2 + 23x + 12 \) is:
\[
(5x + 3)(x + 4)
\]

Would you like more details on any step, or have any questions?

Here are five related questions to explore further:
1. How do you factor a trinomial where the leading coefficient is 1?
2. Can all trinomials be factored using this method?
3. What is the difference between factoring and solving a quadratic equation?
4. How does the quadratic formula relate to factoring?
5. What are the methods for factoring polynomials with four terms?

**Tip:** Always check your factors by expanding them to ensure they multiply back to the original expression!

Learn how to factor the quadratic trinomial 5x^2 + 23x + 12 step-by-step using the trial and error (AC) method. This solution involves factoring techniques, finding common factors, and grouping terms. Suitable for students in Grades 8-10.

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