`(4x + 3y) ^(2) + (4x - 3y) ^(2)` is equal to -

Options

A. `16 x ^(2) - 9y ^(2)`
B. `32 x ^(2) + 18 y ^(2)`
C. ` 16x ^(2) + 9y ^(2)`
D. `32x ^(2) + 9y ^(2)`

Answer

To solve the expression \((4x + 3y)^2 + (4x - 3y)^2\), we can use the algebraic identities for the square of a sum and the square of a difference.

### Step-by-step Solution:

1. **Identify the terms**:
Let \(a = 4x\) and \(b = 3y\). We need to expand \((a + b)^2 + (a - b)^2\).

2. **Use the identity for squares**:
The identity for the square of a sum is:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
The identity for the square of a difference is:
\[ (a - b)^2 = a^2 - 2ab + b^2 \]

3. **Expand both squares**:
- For \((4x + 3y)^2\):
\[ (4x + 3y)^2 = (4x)^2 + 2(4x)(3y) + (3y)^2 = 16x^2 + 24xy + 9y^2 \]
- For \((4x - 3y)^2\):
\[ (4x - 3y)^2 = (4x)^2 - 2(4x)(3y) + (3y)^2 = 16x^2 - 24xy + 9y^2 \]

4. **Combine the results**:
Now, we add the two expansions:
\[ (4x + 3y)^2 + (4x - 3y)^2 = (16x^2 + 24xy + 9y^2) + (16x^2 - 24xy + 9y^2) \]

5. **Simplify the expression**:
Combine like terms:
\[ = 16x^2 + 16x^2 + 24xy - 24xy + 9y^2 + 9y^2 \]
\[ = 32x^2 + 18y^2 \]

### Final Result:
Thus, the expression \((4x + 3y)^2 + (4x - 3y)^2\) simplifies to:
\[ 32x^2 + 18y^2 \]