Subtract `a^(3)+3a-a^(2)-6` from `2a^(2)+a-2a^(3)+3` horizontally.

Answer

To solve the problem of subtracting \( a^3 + 3a - a^2 - 6 \) from \( 2a^2 + a - 2a^3 + 3 \), we will follow these steps:

### Step 1: Write the expression for subtraction
We start with the expression:
\[ (2a^2 + a - 2a^3 + 3) - (a^3 + 3a - a^2 - 6) \]

### Step 2: Distribute the negative sign
When we subtract the second expression, we need to distribute the negative sign across all terms in the parentheses:
\[ = 2a^2 + a - 2a^3 + 3 - a^3 - 3a + a^2 + 6 \]

### Step 3: Combine like terms
Now, we will combine the like terms based on their degrees (the powers of \( a \)):
- For \( a^3 \) terms: \( -2a^3 - a^3 = -3a^3 \)
- For \( a^2 \) terms: \( 2a^2 + a^2 = 3a^2 \)
- For \( a \) terms: \( a - 3a = -2a \)
- For constant terms: \( 3 + 6 = 9 \)

Putting it all together, we have:
\[ -3a^3 + 3a^2 - 2a + 9 \]

### Final Answer
Thus, the result of subtracting \( a^3 + 3a - a^2 - 6 \) from \( 2a^2 + a - 2a^3 + 3 \) is:
\[ -3a^3 + 3a^2 - 2a + 9 \]

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