Sure, let's factor the polynomial [tex]\(2x^3 + 6x^2 + 5x + 15\)[/tex] using the grouping method step-by-step.
### Step 1: Grouping the Polynomial into Two Pairs
First, we group the terms:
[tex]\[
2x^3 + 6x^2 + 5x + 15 = (2x^3 + 6x^2) + (5x + 15)
\][/tex]
### Step 2: Factor Out the GCD (Greatest Common Divisor) from Each Group
Group 1: [tex]\(2x^3 + 6x^2\)[/tex]
In this group, we can factor out the common factor [tex]\(2x^2\)[/tex]:
[tex]\[
2x^3 + 6x^2 = 2x^2(x + 3)
\][/tex]
Group 2: [tex]\(5x + 15\)[/tex]
In this group, we can factor out the common factor [tex]\(5\)[/tex]:
[tex]\[
5x + 15 = 5(x + 3)
\][/tex]
### Step 3: Combine the Factored Terms
Now we have:
[tex]\[
2x^3 + 6x^2 + 5x + 15 = 2x^2(x + 3) + 5(x + 3)
\][/tex]
### Step 4: Factor Out the Common Binomial Factor
Notice that both terms now contain the common binomial factor [tex]\((x + 3)\)[/tex]:
[tex]\[
2x^2(x + 3) + 5(x + 3) = (x + 3)(2x^2 + 5)
\][/tex]
So, the polynomial can be factored completely as:
[tex]\[
2x^3 + 6x^2 + 5x + 15 = (x + 3)(2x^2 + 5)
\][/tex]
### Conclusion
The completely factored form of the polynomial [tex]\(2x^3 + 6x^2 + 5x + 15\)[/tex] is [tex]\((x + 3)(2x^2 + 5)\)[/tex], which corresponds to the correct answer option:
C. [tex]\(\left(2 x^2 + 5\right)(x + 3)\)[/tex]
