Simplify : `(x^(3)-4-x+4x^(2))/(x^(2)+3x-4)`

Options

A. `4+x`
B. `2+x`
C. `1-x`
D. `x+1`

Answer

To simplify the expression \((x^3 - 4 - x + 4x^2)/(x^2 + 3x - 4)\), we can follow these steps:

### Step 1: Arrange the Numerator
First, we need to arrange the numerator in descending order of powers of \(x\):
\[ x^3 + 4x^2 - x - 4 \]

### Step 2: Factor the Denominator
Next, we factor the denominator \(x^2 + 3x - 4\). We look for two numbers that multiply to \(-4\) (the constant term) and add to \(3\) (the coefficient of \(x\)). The numbers \(4\) and \(-1\) fit this requirement:
\[ x^2 + 3x - 4 = (x + 4)(x - 1) \]

### Step 3: Factor the Numerator
Now, we will factor the numerator \(x^3 + 4x^2 - x - 4\). We can use the factor theorem to check for possible rational roots. Testing \(x = -1\):
\[ (-1)^3 + 4(-1)^2 - (-1) - 4 = -1 + 4 + 1 - 4 = 0 \]
Since \(x = -1\) is a root, we can factor \(x + 1\) out of the numerator. We can perform polynomial long division or synthetic division to find the other factor.

#### Performing Synthetic Division:
\[ \begin{array}{r|rrrr} -1 & 1 & 4 & -1 & -4 \\ & & -1 & -3 & 4 \\ \hline & 1 & 3 & -4 & 0 \\ \end{array} \]
The result is \(x^2 + 3x - 4\). Thus, we can write:
\[ x^3 + 4x^2 - x - 4 = (x + 1)(x^2 + 3x - 4) \]

### Step 4: Substitute Back into the Expression
Now we can substitute back into the original expression:
\[ \frac{(x + 1)(x^2 + 3x - 4)}{x^2 + 3x - 4} \]

### Step 5: Cancel Common Factors
Since \(x^2 + 3x - 4\) is present in both the numerator and denominator, we can cancel it out:
\[ x + 1 \]

### Final Answer
Thus, the simplified expression is:
\[ \boxed{x + 1} \]