If `g(x+3)=x^(2)+2x+1`, then which of the following gives g(x)?

Options

A. `g(x)=x^(2)+8x+16`
B. `g(x)=x^(2)-4x+4`
C. `g(x)=x^(2)+2x+4`
D. `g(x)=x^(2)+2x-2`

Explanation

To find \( g(x) \) from the equation \( g(x+3) = x^2 + 2x + 1 \), we will follow these steps:

### Step 1: Substitute \( u \)
Let \( u = x + 3 \). This means that \( x = u - 3 \).

### Step 2: Rewrite the function
Now, we can express \( g(u) \) in terms of \( u \):
\[ g(u) = (u - 3)^2 + 2(u - 3) + 1 \]

### Step 3: Expand the equation
Now, we will expand the right-hand side:
1. Expand \( (u - 3)^2 \):
\[ (u - 3)^2 = u^2 - 6u + 9 \]
2. Expand \( 2(u - 3) \):
\[ 2(u - 3) = 2u - 6 \]
3. Combine all parts:
\[ g(u) = (u^2 - 6u + 9) + (2u - 6) + 1 \]

### Step 4: Combine like terms
Now, combine all the terms:
\[ g(u) = u^2 - 6u + 2u + 9 - 6 + 1 \]
\[ g(u) = u^2 - 4u + 4 \]

### Step 5: Substitute back for \( g(x) \)
Since \( u = x \), we can write:
\[ g(x) = x^2 - 4x + 4 \]

### Final Result
Thus, the function \( g(x) \) is:
\[ g(x) = x^2 - 4x + 4 \]