Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\(4x^{2} + 16x + 12 = 4(x + 1)(x + 3)\)
Solution
To factor the quadratic expression \( 4x^{2} + 16x + 12 \) completely, follow these steps:
1. **Factor Out the Greatest Common Factor (GCF):**
First, identify the GCF of the coefficients \(4\), \(16\), and \(12\), which is \(4\).
\[
4x^{2} + 16x + 12 = 4(x^{2} + 4x + 3)
\]
2. **Factor the Quadratic Inside the Parentheses:**
Now, focus on factoring \(x^{2} + 4x + 3\). We need two numbers that multiply to \(3\) (the constant term) and add up to \(4\) (the coefficient of \(x\)).
The numbers \(1\) and \(3\) satisfy this condition because:
\[
1 \times 3 = 3 \quad \text{and} \quad 1 + 3 = 4
\]
So, we can write:
\[
x^{2} + 4x + 3 = (x + 1)(x + 3)
\]
3. **Combine the Factors:**
Substitute the factored form back into the expression with the GCF:
\[
4(x^{2} + 4x + 3) = 4(x + 1)(x + 3)
\]
**Final Factored Form:**
\[
4x^{2} + 16x + 12 = 4(x + 1)(x + 3)
\]
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Simplify this solution Extra Insights
To factor the quadratic expression \( 4x^{2} + 16x + 12 \), we can start by looking for a common factor. In this case, all terms are divisible by 4. So, we factor out a 4: \[ 4(x^{2} + 4x + 3) \] Now, we need to factor the quadratic \( x^{2} + 4x + 3 \). We look for two numbers that multiply to 3 (the constant term) and add up to 4 (the coefficient of \( x \)). These numbers are 3 and 1. Thus, we can write: \[ x^{2} + 4x + 3 = (x + 1)(x + 3) \] Putting it all together, we have: \[ 4(x + 1)(x + 3) \] So, the completely factored form of \( 4x^{2} + 16x + 12 \) is \( 4(x + 1)(x + 3) \). Now that's some algebra magic!
