To simplify the given expression [tex]\( 3a^2b + 5ab - 4ab + ab^2 + 8a^2b \)[/tex], let's perform the following steps:
1. Combine like terms involving [tex]\( a^2b \)[/tex]:
- We have [tex]\( 3a^2b \)[/tex] and [tex]\( 8a^2b \)[/tex].
- Combining these, we get:
[tex]\[
3a^2b + 8a^2b = 11a^2b
\][/tex]
2. Combine like terms involving [tex]\( ab \)[/tex]:
- We have [tex]\( 5ab \)[/tex] and [tex]\( -4ab \)[/tex].
- Combining these, we get:
[tex]\[
5ab - 4ab = ab
\][/tex]
3. Identify the remaining term [tex]\( ab^2 \)[/tex]:
- We have one term which is [tex]\( ab^2 \)[/tex].
- This term has no other like terms to combine with.
4. Combine all the simplified parts:
- Summarizing our results from the previous steps, we put together all simplified terms:
[tex]\[
11a^2b + ab + ab^2
\][/tex]
5. Factor the simplified expression:
- We notice a common factor of [tex]\( ab \)[/tex] in each term:
[tex]\[
11a^2b + ab + ab^2 = ab(11a + b + 1)
\][/tex]
Thus, the simplified form of the expression [tex]\( 3a^2b + 5ab - 4ab + ab^2 + 8a^2b \)[/tex] is:
[tex]\[
ab(11a + b + 1)
\][/tex]
