Simplify the expression : `5m^(2)n^(2) - 3n^(2) + 6m^(2) - (9n^(2) + 2m^(2)n^(2))` and find its value when m = -2 and n = 4.

Options

A. 192
B. 96
C. 240
D. 24

Answer

To simplify the expression \( 5m^2n^2 - 3n^2 + 6m^2 - (9n^2 + 2m^2n^2) \) and find its value when \( m = -2 \) and \( n = 4 \), we will follow these steps:

### Step 1: Distribute the negative sign
Start by distributing the negative sign across the terms inside the parentheses:

\[ 5m^2n^2 - 3n^2 + 6m^2 - 9n^2 - 2m^2n^2 \]

### Step 2: Combine like terms
Now, we will group and combine the like terms. The like terms here are those that have the same variables raised to the same powers.

- For \( m^2n^2 \):
\[ 5m^2n^2 - 2m^2n^2 = (5 - 2)m^2n^2 = 3m^2n^2 \]

- For \( n^2 \):
\[ -3n^2 - 9n^2 = (-3 - 9)n^2 = -12n^2 \]

- For \( m^2 \):
\[ 6m^2 \]

Putting it all together, we have:

\[ 3m^2n^2 - 12n^2 + 6m^2 \]

### Step 3: Write the simplified expression
Thus, the simplified expression is:

\[ 3m^2n^2 + 6m^2 - 12n^2 \]

### Step 4: Substitute the values of \( m \) and \( n \)
Now, we substitute \( m = -2 \) and \( n = 4 \) into the simplified expression:

\[ 3(-2)^2(4)^2 + 6(-2)^2 - 12(4)^2 \]

### Step 5: Calculate each term
Calculate each term step by step:

1. Calculate \( (-2)^2 = 4 \)
2. Calculate \( (4)^2 = 16 \)

Now substitute these values back into the expression:

\[ 3(4)(16) + 6(4) - 12(16) \]

### Step 6: Perform the multiplications
Now perform the multiplications:

1. \( 3 \times 4 \times 16 = 192 \)
2. \( 6 \times 4 = 24 \)
3. \( 12 \times 16 = 192 \)

### Step 7: Combine the results
Now combine these results:

\[ 192 + 24 - 192 \]

### Step 8: Simplify the final result
Finally, simplify:

\[ 192 - 192 + 24 = 0 + 24 = 24 \]

### Final Answer
The value of the expression when \( m = -2 \) and \( n = 4 \) is:

\[ \boxed{24} \]