To multiply the given polynomials [tex]\( 3x^2 - 5x + 1 \)[/tex] and [tex]\( x^2 + 2x + 4 \)[/tex] using the vertical multiplication method, follow these steps:
### Step-by-Step Process
1. List the Polynomials:
- First Polynomial: [tex]\( 3x^2 - 5x + 1 \)[/tex]
- Second Polynomial: [tex]\( x^2 + 2x + 4 \)[/tex]
2. Multiply Each Term Independently:
- Multiply [tex]\( 3x^2 \)[/tex] by Each Term in [tex]\( x^2 + 2x + 4 \)[/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]
- Multiply [tex]\(-5x\)[/tex] by Each Term in [tex]\( x^2 + 2x + 4 \)[/tex]:
- [tex]\(-5x \cdot x^2 = -5x^3 \)[/tex]
- [tex]\(-5x \cdot 2x = -10x^2 \)[/tex]
- [tex]\(-5x \cdot 4 = -20x \)[/tex]
- Multiply [tex]\( 1 \)[/tex] by Each Term in [tex]\( x^2 + 2x + 4 \)[/tex]:
- [tex]\( 1 \cdot x^2 = x^2 \)[/tex]
- [tex]\( 1 \cdot 2x = 2x \)[/tex]
- [tex]\( 1 \cdot 4 = 4 \)[/tex]
3. Combine All the Results:
- [tex]\( 3x^4 \)[/tex]
- [tex]\( 6x^3 + (-5x^3) = x^3 \)[/tex]
- [tex]\( 12x^2 + (-10x^2) + x^2 = 3x^2 \)[/tex]
- [tex]\(-20x + 2x = -18x \)[/tex]
- [tex]\( 4 \)[/tex]
4. Sum the Combined Results:
- [tex]\( 3x^4 + x^3 + 3x^2 - 18x + 4 \)[/tex]
### Conclusion
Therefore, the final result of multiplying [tex]\( 3x^2 - 5x + 1 \)[/tex] by [tex]\( x^2 + 2x + 4 \)[/tex] using the vertical multiplication method is:
[tex]\[ 3x^4 + x^3 + 3x^2 - 18x + 4 \][/tex]
Thus, the correct answer is:
C. [tex]\( 3x^4 + x^3 + 3x^2 - 18x + 4 \)[/tex]
