Subtract `3x^(3) - 5x^(2) - 9x + 6 " from " 2x^(3) + 3y^(2) - 4z^(2) and x^(2) - 2y^(2) +z^(2)`

Answer

To solve the problem of subtracting \(3x^3 - 5x^2 - 9x + 6\) from \(2x^3 + 3y^2 - 4z^2\) and \(x^2 - 2y^2 + z^2\), we will follow these steps:

### Step 1: Write down the expressions
We have two expressions from which we need to subtract the polynomial \(3x^3 - 5x^2 - 9x + 6\):
1. \(2x^3 + 3y^2 - 4z^2\)
2. \(x^2 - 2y^2 + z^2\)

### Step 2: Subtract from the first expression
We start by subtracting \(3x^3 - 5x^2 - 9x + 6\) from \(2x^3 + 3y^2 - 4z^2\):
\[ (2x^3 + 3y^2 - 4z^2) - (3x^3 - 5x^2 - 9x + 6) \]

### Step 3: Distribute the negative sign
Distributing the negative sign gives us:
\[ 2x^3 + 3y^2 - 4z^2 - 3x^3 + 5x^2 + 9x - 6 \]

### Step 4: Combine like terms
Now we combine the like terms:
- For \(x^3\): \(2x^3 - 3x^3 = -x^3\)
- For \(x^2\): \(5x^2\) (only term)
- For \(x\): \(9x\) (only term)
- For \(y^2\): \(3y^2\) (only term)
- For \(z^2\): \(-4z^2\) (only term)
- Constant term: \(-6\)

Putting it all together, we get:
\[ -x^3 + 5x^2 + 9x + 3y^2 - 4z^2 - 6 \]

### Step 5: Subtract from the second expression
Next, we subtract \(3x^3 - 5x^2 - 9x + 6\) from \(x^2 - 2y^2 + z^2\):
\[ (x^2 - 2y^2 + z^2) - (3x^3 - 5x^2 - 9x + 6) \]

### Step 6: Distribute the negative sign
Distributing the negative sign gives us:
\[ x^2 - 2y^2 + z^2 - 3x^3 + 5x^2 + 9x - 6 \]

### Step 7: Combine like terms
Now we combine the like terms:
- For \(x^3\): \(-3x^3\) (only term)
- For \(x^2\): \(x^2 + 5x^2 = 6x^2\)
- For \(y^2\): \(-2y^2\) (only term)
- For \(z^2\): \(z^2\) (only term)
- For \(x\): \(9x\) (only term)
- Constant term: \(-6\)

Putting it all together, we get:
\[ -3x^3 + 6x^2 + 9x - 2y^2 + z^2 - 6 \]

### Final Solutions
1. From the first expression: \(-x^3 + 5x^2 + 9x + 3y^2 - 4z^2 - 6\)
2. From the second expression: \(-3x^3 + 6x^2 + 9x - 2y^2 + z^2 - 6\)