`(z^(2)+13)(z^(2)-5)=` ____

Options

A. `2z^(4)+18z^(2)-8`
B. `z^(4)+8z^(2)-65`
C. `z^(4)-8z^(2)-65`
D. `z^(4)+8z^(2)+65`

Answer

To solve the expression \((z^{2}+13)(z^{2}-5)\), we will use the distributive property (also known as the FOIL method for binomials). Let's break it down step by step.

### Step-by-Step Solution:

1. **Write the expression**:
\[ (z^{2}+13)(z^{2}-5) \]

2. **Distribute each term in the first bracket with each term in the second bracket**:
- Multiply \(z^{2}\) by \(z^{2}\):
\[ z^{2} \cdot z^{2} = z^{4} \]
- Multiply \(z^{2}\) by \(-5\):
\[ z^{2} \cdot (-5) = -5z^{2} \]
- Multiply \(13\) by \(z^{2}\):
\[ 13 \cdot z^{2} = 13z^{2} \]
- Multiply \(13\) by \(-5\):
\[ 13 \cdot (-5) = -65 \]

3. **Combine all the results**:
\[ z^{4} - 5z^{2} + 13z^{2} - 65 \]

4. **Combine like terms**:
- Combine \(-5z^{2}\) and \(13z^{2}\):
\[ -5z^{2} + 13z^{2} = 8z^{2} \]
- So, the expression simplifies to:
\[ z^{4} + 8z^{2} - 65 \]

5. **Final result**:
\[ (z^{2}+13)(z^{2}-5) = z^{4} + 8z^{2} - 65 \]

### Final Answer:
\[ (z^{2}+13)(z^{2}-5) = z^{4} + 8z^{2} - 65 \]