If `g(x-1) = x^(2)+2` , then g(x)=

Options

A. `x^(2)-2x + 3`
B. `x^(2) + 2x +3`
C. `x^(2)-3x+2`
D. `x^(2)+2`

Explanation

To find \( g(x) \) given that \( g(x-1) = x^2 + 2 \), we can follow these steps:

### Step 1: Rewrite \( g(x) \)
We know that \( g(x-1) = x^2 + 2 \). To express \( g(x) \), we can substitute \( x \) with \( x + 1 \) in the equation. This means we need to find \( g(x) \) by replacing \( x-1 \) with \( x \):
\[ g(x) = g((x + 1) - 1) = (x + 1)^2 + 2 \]

### Step 2: Expand the expression
Now we will expand \( (x + 1)^2 + 2 \):
\[ g(x) = (x + 1)^2 + 2 \]
Using the identity \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[ g(x) = x^2 + 2x + 1 + 2 \]

### Step 3: Combine like terms
Now, we will combine the constant terms:
\[ g(x) = x^2 + 2x + 3 \]

Thus, the final expression for \( g(x) \) is:
\[ \boxed{x^2 + 2x + 3} \]

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