`f(x)=(x^(4)+2x^(3)-3x^(2)+4x+12)/(x+3)`
Which of the following functions is equivalent to the functions above for all values of x for which function above for all values of x for which function f is defined?

Options

A. `g(x)=x^(3)-x^(2)+4`
B. `g(x)=x^(2)-x+4`
C. `g(x)=x^(3)-x^(2)+4x`
D. `g(x)=x^(4)+2x^(3)-3x^(2)+4`

Answer

To solve the given problem, we need to simplify the function \( f(x) = \frac{x^4 + 2x^3 - 3x^2 + 4x + 12}{x + 3} \) by performing polynomial long division.

### Step-by-Step Solution:

1. **Set Up the Division**:
We want to divide the polynomial \( x^4 + 2x^3 - 3x^2 + 4x + 12 \) by \( x + 3 \).

2. **Divide the Leading Terms**:
Divide the leading term of the numerator \( x^4 \) by the leading term of the denominator \( x \):
\[ x^4 \div x = x^3 \]

3. **Multiply and Subtract**:
Multiply \( x^3 \) by \( x + 3 \):
\[ x^3(x + 3) = x^4 + 3x^3 \]
Now subtract this from the original polynomial:
\[ (x^4 + 2x^3 - 3x^2 + 4x + 12) - (x^4 + 3x^3) = -x^3 - 3x^2 + 4x + 12 \]

4. **Repeat the Process**:
Now, take the new leading term \( -x^3 \) and divide it by \( x \):
\[ -x^3 \div x = -x^2 \]
Multiply \( -x^2 \) by \( x + 3 \):
\[ -x^2(x + 3) = -x^3 - 3x^2 \]
Subtract this from the previous result:
\[ (-x^3 - 3x^2 + 4x + 12) - (-x^3 - 3x^2) = 4x + 12 \]

5. **Continue Dividing**:
Now take \( 4x \) and divide it by \( x \):
\[ 4x \div x = 4 \]
Multiply \( 4 \) by \( x + 3 \):
\[ 4(x + 3) = 4x + 12 \]
Subtract this from the previous result:
\[ (4x + 12) - (4x + 12) = 0 \]

6. **Conclusion**:
Since the remainder is \( 0 \), we conclude that:
\[ f(x) = x^3 - x^2 + 4 \]
This means that the function \( f(x) \) is equivalent to \( x^3 - x^2 + 4 \) for all values of \( x \) for which \( f \) is defined.

### Final Answer:
The equivalent function is \( x^3 - x^2 + 4 \).