Final answer:
To expand the given polynomial, we distribute each term inside the parentheses and then combine like terms, resulting in the simplified polynomial ax^2 - 3a^2x + 5x^3.
Step-by-step explanation:
To expand the polynomial ax(2x - 3a) - x(ax - 5x^2), we apply the distributive property (also known as the foil method) to each term. Let's break it down step-by-step:
- Multiply ax by 2x to get 2ax^2.
- Multiply ax by -3a to get -3a^2x.
- Multiply -x by ax to get -ax^2.
- Multiply -x by -5x^2 to get +5x^3.
Now combine these results:
2ax^2 - 3a^2x - ax^2 + 5x^3
Combine like terms to simplify further:
(2ax^2 - ax^2) - 3a^2x + 5x^3
This simplifies to:
ax^2 - 3a^2x + 5x^3
we have:
ax(2x - 3a) = 2ax^2 - 3a^2x
Next, we simplify the second term:
x(ax - 5x^2) = ax^2 - 5x^3
Combining both terms, we get:
2ax^2 - 3a^2x - ax^2 + 5x^3
Finally, we can combine like terms:
(2a - 1)x^2 - 3a^2x + 5x^3