Sure! Let's factor the given polynomial step by step:
Given the polynomial:
[tex]\[ 2x^2 + 8x + 6 \][/tex]
### Step 1: Identify the coefficients.
Here, the coefficients are:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 8 \)[/tex]
- [tex]\( c = 6 \)[/tex]
### Step 2: Consider factoring out the greatest common factor (if any).
In this case, there isn't a greatest common factor other than 1.
### Step 3: Express the polynomial in a suitable form for factoring.
Let's rewrite the quadratic expression:
[tex]\[ 2x^2 + 8x + 6 = 2(x^2 + 4x + 3) \][/tex]
### Step 4: Focus on the quadratic expression inside the parentheses.
We are now working with:
[tex]\[ x^2 + 4x + 3 \][/tex]
### Step 5: Find two numbers that multiply to the constant term (3) and add up to the middle coefficient (4).
The numbers that meet these criteria are 1 and 3, because:
[tex]\[ 1 \times 3 = 3 \][/tex]
[tex]\[ 1 + 3 = 4 \][/tex]
### Step 6: Write the quadratic as a product of binomials.
Thus,
[tex]\[ x^2 + 4x + 3 = (x + 1)(x + 3) \][/tex]
### Step 7: Incorporate the factor that was factored out initially.
Now, include the factor of 2:
[tex]\[ 2(x + 1)(x + 3) \][/tex]
Therefore, the completely factored form of the polynomial [tex]\( 2x^2 + 8x + 6 \)[/tex] is:
[tex]\[ 2(x + 1)(x + 3) \][/tex]
So, the correct answer is:
[tex]\[ \boxed{2(x + 3)(x + 1)} \][/tex]
