To factor the quadratic expression \(x^2 + 8x + 12\) by splitting the middle term, follow these steps:

### Step 1: Identify the middle term and constant term.
- Middle term: \(8x\)
- Constant term: \(12\)

### Step 2: Find two numbers that multiply to the constant term (\(12\)) and add to the coefficient of the middle term (\(8\)).

We need two numbers whose product is \(12\) and whose sum is \(8\). The numbers \(2\) and \(6\) satisfy this requirement because:
\[
2 \times 6 = 12 \quad \text{and} \quad 2 + 6 = 8
\]

### Step 3: Split the middle term using these two numbers.
We can rewrite the quadratic expression as:
\[
x^2 + 2x + 6x + 12
\]

### Step 4: Factor by grouping.
Group the terms in pairs:
\[
(x^2 + 2x) + (6x + 12)
\]

Now, factor out the greatest common factor (GCF) from each group:
\[
x(x + 2) + 6(x + 2)
\]

### Step 5: Factor out the common binomial factor.
Notice that both groups contain \((x + 2)\), so we factor that out:
\[
(x + 2)(x + 6)
\]

### Final Answer:
The expression \(x^2 + 8x + 12\) factors to \((x + 2)(x + 6)\).

Would you like to go over any of these steps in more detail or have any other questions?

Here are five related questions to consider:
1. How can you determine if a quadratic expression is factorable by integers?
2. What does the factorization of \(x^2 + 8x + 12\) tell you about the roots of the corresponding equation?
3. How does factoring by grouping work with different types of quadratic expressions?
4. What would happen if the quadratic had no real number solutions? How would the factorization process change?
5. How can the quadratic formula be used as a check after factoring a quadratic expression?

**Tip:** When factoring quadratics, always look for pairs of factors of the constant term that add up to the middle coefficient. This approach simplifies the factoring process.

Learn how to factor the quadratic expression x^2 + 8x + 12 using the method of splitting the middle term. Follow a detailed step-by-step solution that explains how to identify the middle term and constant term, find suitable factors, and factorize by grouping. Suitable for students in Grades 7-9 studying algebra and quadratic equations.

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