What should be subtacted from `7a^(3)+6a^(2)-2-5a` to get `4a^(3)-7a+5a^(2)+8`?

Options

A. `3a^(3)+a^(2)+2a-10`
B. `4a^(3)-a^(2)-2a+8`
C. `-2a^(3)-a^(2)+3a-4`
D. `3a^(3)-2a^(2)+a-5`

Answer

To solve the problem of what should be subtracted from \(7a^3 + 6a^2 - 2 - 5a\) to get \(4a^3 - 7a + 5a^2 + 8\), we can follow these steps:

### Step 1: Set up the equation
We want to find \(x\) such that:
\[ 7a^3 + 6a^2 - 2 - 5a - x = 4a^3 - 7a + 5a^2 + 8 \]

### Step 2: Rearrange the equation
To find \(x\), we can rearrange the equation:
\[ x = 7a^3 + 6a^2 - 2 - 5a - (4a^3 - 7a + 5a^2 + 8) \]

### Step 3: Distribute the negative sign
Now, distribute the negative sign across the terms in the parentheses:
\[ x = 7a^3 + 6a^2 - 2 - 5a - 4a^3 + 7a - 5a^2 - 8 \]

### Step 4: Combine like terms
Now, combine the like terms:
- For \(a^3\): \(7a^3 - 4a^3 = 3a^3\)
- For \(a^2\): \(6a^2 - 5a^2 = 1a^2\)
- For \(a\): \(-5a + 7a = 2a\)
- For the constant: \(-2 - 8 = -10\)

Putting it all together, we get:
\[ x = 3a^3 + 1a^2 + 2a - 10 \]

### Final Answer
Thus, what should be subtracted from \(7a^3 + 6a^2 - 2 - 5a\) to get \(4a^3 - 7a + 5a^2 + 8\) is:
\[ 3a^3 + a^2 + 2a - 10 \]

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