Subtract the sum of `10m - 8n + 6p^(2) "and" - 7m - 4n + 12p^(2)` from the sum `2m - 4n + 6p^(2) "and" m-8n + 12p^(2)`.

Answer

To solve the problem, we need to follow these steps:

### Step 1: Find the sum of the first two expressions
We need to calculate the sum of \(10m - 8n + 6p^2\) and \(-7m - 4n + 12p^2\).

\[ (10m - 8n + 6p^2) + (-7m - 4n + 12p^2) \]

Combine like terms:

- For \(m\): \(10m - 7m = 3m\)
- For \(n\): \(-8n - 4n = -12n\)
- For \(p^2\): \(6p^2 + 12p^2 = 18p^2\)

So, the sum is:

\[ 3m - 12n + 18p^2 \quad \text{(Equation 1)} \]

### Step 2: Find the sum of the next two expressions
Now, we need to calculate the sum of \(2m - 4n + 6p^2\) and \(m - 8n + 12p^2\).

\[ (2m - 4n + 6p^2) + (m - 8n + 12p^2) \]

Combine like terms:

- For \(m\): \(2m + m = 3m\)
- For \(n\): \(-4n - 8n = -12n\)
- For \(p^2\): \(6p^2 + 12p^2 = 18p^2\)

So, the sum is:

\[ 3m - 12n + 18p^2 \quad \text{(Equation 2)} \]

### Step 3: Subtract Equation 1 from Equation 2
Now we need to subtract the result from Equation 1 from the result in Equation 2.

\[ (3m - 12n + 18p^2) - (3m - 12n + 18p^2) \]

Distributing the negative sign:

\[ 3m - 12n + 18p^2 - 3m + 12n - 18p^2 \]

### Step 4: Combine like terms
Now, combine the terms:

- For \(m\): \(3m - 3m = 0\)
- For \(n\): \(-12n + 12n = 0\)
- For \(p^2\): \(18p^2 - 18p^2 = 0\)

So, the final result is:

\[ 0 \]

### Final Answer
The final answer is \(0\).

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