To solve the expression [tex]\((3a^6 + 4b^5)(5ab + 7b)\)[/tex], follow these steps:
1. Distribute each term in the first polynomial by each term in the second polynomial.
2. Multiply each pair of terms:
- First, distribute [tex]\(3a^6\)[/tex] to both [tex]\(5ab\)[/tex] and [tex]\(7b\)[/tex]:
- [tex]\(3a^6 \cdot 5ab = 15a^7b\)[/tex]
- [tex]\(3a^6 \cdot 7b = 21a^6b\)[/tex]
- Next, distribute [tex]\(4b^5\)[/tex] to both [tex]\(5ab\)[/tex] and [tex]\(7b\)[/tex]:
- [tex]\(4b^5 \cdot 5ab = 20ab^6\)[/tex]
- [tex]\(4b^5 \cdot 7b = 28b^6\)[/tex]
3. Combine all the terms after distribution to get the expanded expression:
[tex]\[
15a^7b + 21a^6b + 20ab^6 + 28b^6
\][/tex]
Thus, the expansion of [tex]\((3a^6 + 4b^5)(5ab + 7b)\)[/tex] is
[tex]\[
15a^7b + 21a^6b + 20ab^6 + 28b^6
\][/tex]
