Sure, let's go through the step-by-step process to multiply the polynomials [tex]\( (3x^2 - 5x + 1) \)[/tex] and [tex]\( (x^2 + 2x + 4) \)[/tex] using the vertical multiplication method.
### Step 1: Write down the polynomials
[tex]\[
(3x^2 - 5x + 1) \quad \text{and} \quad (x^2 + 2x + 4)
\][/tex]
### Step 2: Multiply each term of the first polynomial with each term of the second polynomial
1. Multiply the leading term [tex]\(3x^2\)[/tex]:
- [tex]\(3x^2 \cdot x^2 = 3x^4\)[/tex]
- [tex]\(3x^2 \cdot 2x = 6x^3\)[/tex]
- [tex]\(3x^2 \cdot 4 = 12x^2\)[/tex]
2. Multiply the middle term [tex]\(-5x:
- \(-5x \cdot x^2 = -5x^3\)[/tex]
- [tex]\(-5x \cdot 2x = -10x^2\)[/tex]
- [tex]\(-5x \cdot 4 = -20x\)[/tex]
3. Multiply the constant term [tex]\(1\)[/tex]:
- [tex]\(1 \cdot x^2 = x^2\)[/tex]
- [tex]\(1 \cdot 2x = 2x\)[/tex]
- [tex]\(1 \cdot 4 = 4\)[/tex]
### Step 3: Write down all the terms obtained from the multiplication
[tex]\[
3x^4, \quad 6x^3, \quad 12x^2, \quad -5x^3, \quad -10x^2, \quad -20x, \quad x^2, \quad 2x, \quad 4
\][/tex]
### Step 4: Combine like terms
- [tex]\(x^4\)[/tex] terms: [tex]\(3x^4\)[/tex]
- [tex]\(x^3\)[/tex] terms: [tex]\(6x^3 - 5x^3 = x^3\)[/tex]
- [tex]\(x^2\)[/tex] terms: [tex]\(12x^2 - 10x^2 + x^2 = 3x^2\)[/tex]
- [tex]\(x\)[/tex] terms: [tex]\(-20x + 2x = -18x\)[/tex]
- Constant term: [tex]\(4\)[/tex]
So, combining all like terms, we get:
[tex]\[
3x^4 + x^3 + 3x^2 - 18x + 4
\][/tex]
### Step 5: Choose the correct answer
The final expression matches option C.
So the correct answer is:
[tex]\[
\boxed{C. \; 3x^4 + x^3 + 3x^2 - 18x + 4}
\][/tex]
