To simplify the expression [tex]\((2a - 3b) \left(a^2 - ab + 3b\right)\)[/tex], follow these steps:
1. Expand the expression using distribution:
Multiply each term in the first polynomial [tex]\((2a - 3b)\)[/tex] by each term in the second polynomial [tex]\((a^2 - ab + 3b)\)[/tex]:
[tex]\[
(2a - 3b)(a^2 - ab + 3b) = 2a \cdot a^2 + 2a \cdot (-ab) + 2a \cdot 3b - 3b \cdot a^2 - 3b \cdot (-ab) - 3b \cdot 3b
\][/tex]
2. Carry out the individual multiplications:
- For [tex]\(2a \cdot a^2\)[/tex], the result is [tex]\(2a^3\)[/tex].
- For [tex]\(2a \cdot (-ab)\)[/tex], the result is [tex]\(-2a^2b\)[/tex].
- For [tex]\(2a \cdot 3b\)[/tex], the result is [tex]\(6ab\)[/tex].
- For [tex]\(-3b \cdot a^2\)[/tex], the result is [tex]\(-3a^2b\)[/tex].
- For [tex]\(-3b \cdot (-ab)\)[/tex], the result is [tex]\(3ab^2\)[/tex].
- For [tex]\(-3b \cdot 3b\)[/tex], the result is [tex]\(-9b^2\)[/tex].
3. Combine all these results:
[tex]\[
2a^3 - 2a^2b + 6ab - 3a^2b + 3ab^2 - 9b^2
\][/tex]
4. Combine like terms:
- Combine [tex]\(2a^2b\)[/tex] and [tex]\(-3a^2b\)[/tex] which results in [tex]\(-5a^2b\)[/tex].
- The remaining terms are already isolated or do not require further simplification.
Therefore, combining all results, we get:
[tex]\[
2a^3 - 5a^2b + 3ab^2 + 6ab - 9b^2
\][/tex]
The simplified expression is [tex]\(2a^3 - 5a^2b + 3ab^2 + 6ab - 9b^2\)[/tex].
