To find the product of [tex]\((a^3 + 2b^3 - 3c + 1)\)[/tex] and [tex]\((a^2 + 2b + 3)\)[/tex], we need to distribute each term in the first polynomial by each term in the second polynomial and then combine like terms. Here is a step-by-step solution:
1. Distribute each term in the first polynomial by each term in the second polynomial:
- [tex]\(a^3 (a^2 + 2b + 3)\)[/tex]
- [tex]\(2b^3 (a^2 + 2b + 3)\)[/tex]
- [tex]\(-3c (a^2 + 2b + 3)\)[/tex]
- [tex]\(1 (a^2 + 2b + 3)\)[/tex]
2. Calculate each of these distributions separately:
- [tex]\(a^3 \cdot a^2 = a^5\)[/tex]
- [tex]\(a^3 \cdot 2b = 2a^3 b\)[/tex]
- [tex]\(a^3 \cdot 3 = 3a^3\)[/tex]
- [tex]\(2b^3 \cdot a^2 = 2a^2 b^3\)[/tex]
- [tex]\(2b^3 \cdot 2b = 4b^4\)[/tex]
- [tex]\(2b^3 \cdot 3 = 6b^3\)[/tex]
- [tex]\(-3c \cdot a^2 = -3a^2 c\)[/tex]
- [tex]\(-3c \cdot 2b = -6b c\)[/tex]
- [tex]\(-3c \cdot 3 = -9c\)[/tex]
- [tex]\(1 \cdot a^2 = a^2\)[/tex]
- [tex]\(1 \cdot 2b = 2b\)[/tex]
- [tex]\(1 \cdot 3 = 3\)[/tex]
3. Combine all these results:
- [tex]\(a^5\)[/tex]
- [tex]\(+ 2a^3 b\)[/tex]
- [tex]\(+ 3a^3\)[/tex]
- [tex]\(+ 2a^2 b^3\)[/tex]
- [tex]\(+ 4b^4\)[/tex]
- [tex]\(+ 6b^3\)[/tex]
- [tex]\(- 3a^2 c\)[/tex]
- [tex]\(- 6b c\)[/tex]
- [tex]\(- 9c\)[/tex]
- [tex]\(+ a^2\)[/tex]
- [tex]\(+ 2b\)[/tex]
- [tex]\(+ 3\)[/tex]
4. Combine like terms to get the final polynomial:
[tex]\[
a^5 + 2a^3b + 3a^3 + 2a^2b^3 - 3a^2c + a^2 + 4b^4 + 6b^3 - 6bc + 2b - 9c + 3
\][/tex]
So, the product of [tex]\((a^3 + 2b^3 - 3c + 1)\)[/tex] and [tex]\((a^2 + 2b + 3)\)[/tex] is:
[tex]\[
a^5 + 2a^3b + 3a^3 + 2a^2b^3 - 3a^2c + a^2 + 4b^4 + 6b^3 - 6bc + 2b - 9c + 3
\][/tex]
