Subtract :
`4p^(2) + 5q^(2) - 6r^(2) + 7` from `3p^(2) - 4q^(2) - 5r^(2) - 6`

Answer

To solve the problem of subtracting the expression \( 4p^2 + 5q^2 - 6r^2 + 7 \) from \( 3p^2 - 4q^2 - 5r^2 - 6 \), we can follow these steps:

### Step 1: Write the expression for subtraction
We need to subtract the first expression from the second. This can be written as:
\[ (3p^2 - 4q^2 - 5r^2 - 6) - (4p^2 + 5q^2 - 6r^2 + 7) \]

### Step 2: Distribute the negative sign
When we subtract, we need to distribute the negative sign across the second expression:
\[ 3p^2 - 4q^2 - 5r^2 - 6 - 4p^2 - 5q^2 + 6r^2 - 7 \]

### Step 3: Combine like terms
Now, we will combine the like terms:
- For \( p^2 \) terms: \( 3p^2 - 4p^2 = -1p^2 \) or simply \( -p^2 \)
- For \( q^2 \) terms: \( -4q^2 - 5q^2 = -9q^2 \)
- For \( r^2 \) terms: \( -5r^2 + 6r^2 = 1r^2 \) or simply \( r^2 \)
- For the constant terms: \( -6 - 7 = -13 \)

### Step 4: Write the final expression
Putting all the combined terms together, we get:
\[ -p^2 - 9q^2 + r^2 - 13 \]

### Final Answer
Thus, the result of the subtraction is:
\[ -p^2 - 9q^2 + r^2 - 13 \]

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