Subtract `4x^(2) - z^(2)` from the sum of `2x^(2) + 3y^(2) - 4z^(2) and x^(2) - 2y^(2) + z^(2)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Find the sum of the two expressions
We need to add the expressions \(2x^2 + 3y^2 - 4z^2\) and \(x^2 - 2y^2 + z^2\).

**Calculation:**
\[ (2x^2 + 3y^2 - 4z^2) + (x^2 - 2y^2 + z^2) \]

### Step 2: Combine like terms
Now, we will combine the like terms from the sum we calculated in Step 1.

- For \(x^2\): \(2x^2 + x^2 = 3x^2\)
- For \(y^2\): \(3y^2 - 2y^2 = 1y^2\) or simply \(y^2\)
- For \(z^2\): \(-4z^2 + z^2 = -3z^2\)

So, the sum becomes:
\[ 3x^2 + y^2 - 3z^2 \]

### Step 3: Subtract \(4x^2 - z^2\) from the sum
Next, we need to subtract the expression \(4x^2 - z^2\) from the sum we found in Step 2.

**Calculation:**
\[ (3x^2 + y^2 - 3z^2) - (4x^2 - z^2) \]

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

### Step 5: Combine like terms again
Now, we will combine the like terms again:

- For \(x^2\): \(3x^2 - 4x^2 = -1x^2\) or simply \(-x^2\)
- For \(y^2\): \(y^2\) remains as it is.
- For \(z^2\): \(-3z^2 + z^2 = -2z^2\)

So, the final result is:
\[ -x^2 + y^2 - 2z^2 \]

### Final Answer:
The final answer is:
\[ -x^2 + y^2 - 2z^2 \]

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