To factor the trinomial \( 2x^2 + 8x + 6 \) completely, let's go through the steps:
1. Identify the coefficients:
The trinomial is in the standard form \( ax^2 + bx + c \) where \( a = 2 \), \( b = 8 \), and \( c = 6 \).
2. Look for the greatest common factor (GCF):
First, check if there is a GCF among all the terms. Here, the GCF is 2. Factor out the GCF:
[tex]\[
2x^2 + 8x + 6 = 2(x^2 + 4x + 3)
\][/tex]
3. Factor the quadratic expression inside the parentheses:
Now, focus on factoring \( x^2 + 4x + 3 \).
- Find two numbers that multiply to the constant term \( 3 \) and add up to the linear coefficient \( 4 \).
- These numbers are \( 1 \) and \( 3 \), because \( 1 \times 3 = 3 \) and \( 1 + 3 = 4 \).
4. Write the expression as a product of binomials:
Rewrite \( x^2 + 4x + 3 \) as:
[tex]\[
x^2 + 4x + 3 = (x + 1)(x + 3)
\][/tex]
5. Combine with the GCF:
Now, include the GCF we factored out earlier:
[tex]\[
2(x^2 + 4x + 3) = 2(x + 1)(x + 3)
\][/tex]
Thus, the trinomial \( 2x^2 + 8x + 6 \) factors completely as:
[tex]\[
2(x + 1)(x + 3)
\][/tex]
Answer choice: [tex]\( \boxed{C} \)[/tex] [tex]\( 2(x + 3)(x + 1) \)[/tex] ───────────────────────────────────────────────────────
