If `f(x+2)=(x+3)^(2)-2x`, then `f(x)=`

Options

A. `x^(2)-2`
B. `x^(2)+5`
C. `x^(2)+4x+9`
D. `(x+5)^(2)-(x+2)`

Answer

To solve the problem, we need to find the function \( f(x) \) given that \( f(x + 2) = (x + 3)^2 - 2x \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ f(x + 2) = (x + 3)^2 - 2x \]

2. **Expand the right-hand side:**
\[ (x + 3)^2 = x^2 + 6x + 9 \]
Therefore,
\[ f(x + 2) = x^2 + 6x + 9 - 2x \]

3. **Combine like terms:**
\[ f(x + 2) = x^2 + 4x + 9 \]

4. **Rewrite the expression:**
We can express \( f(x + 2) \) in a different form:
\[ f(x + 2) = x^2 + 4x + 4 + 5 = (x + 2)^2 + 5 \]

5. **Substitute \( x \) with \( x - 2 \):**
To find \( f(x) \), we replace \( x \) with \( x - 2 \) in the expression we derived:
\[ f(x) = ((x - 2) + 2)^2 + 5 = (x)^2 + 5 \]

6. **Final Result:**
Thus, we have:
\[ f(x) = x^2 + 5 \]

### Final Answer:
\[ f(x) = x^2 + 5 \]