Let's work through Problem B step by step, using the same methods as we did for Problem A.
### Problem B
Given:
[tex]\[ 2a^2b(b^3 - 4ab + 5a^2) \][/tex]
We need to multiply each term inside the parenthesis by the monomial [tex]\(2a^2b\)[/tex].
Here's the process in detail:
1. Distribute [tex]\(2a^2b\)[/tex] to each term in the polynomial [tex]\(b^3 - 4ab + 5a^2\)[/tex]:
[tex]\[
2a^2b(b^3 - 4ab + 5a^2) = (2a^2b \cdot b^3) + (2a^2b \cdot -4ab) + (2a^2b \cdot 5a^2)
\][/tex]
2. Multiply [tex]\(2a^2b\)[/tex] by each term individually:
- First term: [tex]\(2a^2b \cdot b^3\)[/tex]
[tex]\[
2a^2b \cdot b^3 = 2a^2b^{1+3} = 2a^2b^4
\][/tex]
- Second term: [tex]\(2a^2b \cdot -4ab\)[/tex]
[tex]\[
2a^2b \cdot -4ab = 2 \cdot -4 \cdot a^2 \cdot a \cdot b \cdot b = -8a^{2+1}b^{1+1} = -8a^3b^2
\][/tex]
- Third term: [tex]\(2a^2b \cdot 5a^2\)[/tex]
[tex]\[
2a^2b \cdot 5a^2 = 2 \cdot 5 \cdot a^2 \cdot a^2 \cdot b = 10a^{2+2}b = 10a^4b
\][/tex]
3. Write the final expression in standard form:
[tex]\[
2a^2b(b^3 - 4ab + 5a^2) = 2a^2b^4 - 8a^3b^2 + 10a^4b
\][/tex]
So the answer for Problem B is:
[tex]\[
2a^2b^4 - 8a^3b^2 + 10a^4b
\][/tex]
