`(x^(2) -x-2)/(2x^(2) +x-3)` in lowest term is:

Options

A. `(x-1)/(2x+3)`
B. `(x^(2) -x-2)/(2x^(2) +x-3)`
C. `(x-2)/(2x+3)`
D. `(x+1)/(x-1)`

Answer

To simplify the expression \(\frac{x^2 - x - 2}{2x^2 + x - 3}\) to its lowest terms, we will factor both the numerator and the denominator.

### Step 1: Factor the Numerator
The numerator is \(x^2 - x - 2\). We need to factor this quadratic expression.

1. We look for two numbers that multiply to \(-2\) (the constant term) and add to \(-1\) (the coefficient of \(x\)).
2. The numbers \(-2\) and \(1\) satisfy these conditions because:
- \(-2 \times 1 = -2\)
- \(-2 + 1 = -1\)

Thus, we can write:
\[ x^2 - x - 2 = (x - 2)(x + 1) \]

### Step 2: Factor the Denominator
The denominator is \(2x^2 + x - 3\). We will factor this quadratic expression as well.

1. We look for two numbers that multiply to \(2 \times -3 = -6\) and add to \(1\) (the coefficient of \(x\)).
2. The numbers \(3\) and \(-2\) satisfy these conditions because:
- \(3 \times -2 = -6\)
- \(3 + (-2) = 1\)

We can rewrite the middle term:
\[ 2x^2 + 3x - 2x - 3 \]
Now we group the terms:
\[ (2x^2 + 3x) + (-2x - 3) \]
Factoring by grouping:
\[ x(2x + 3) - 1(2x + 3) \]
Now we factor out the common term \((2x + 3)\):
\[ 2x^2 + x - 3 = (2x + 3)(x - 1) \]

### Step 3: Rewrite the Expression
Now we can rewrite the original expression using the factored forms:
\[ \frac{x^2 - x - 2}{2x^2 + x - 3} = \frac{(x - 2)(x + 1)}{(2x + 3)(x - 1)} \]

### Step 4: Simplify the Expression
Now we check if there are any common factors in the numerator and denominator.

- The numerator has factors \((x - 2)\) and \((x + 1)\).
- The denominator has factors \((2x + 3)\) and \((x - 1)\).

Since there are no common factors, the expression is already in its simplest form.

### Final Result
Thus, the expression \(\frac{x^2 - x - 2}{2x^2 + x - 3}\) in lowest terms is:
\[ \frac{(x - 2)(x + 1)}{(2x + 3)(x - 1)} \]