Express the following in the lowest terms.
`((x-3)(x^(2)-5x+4))/((x-4)(x^(2)-2x-3)):`

Options

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

Answer

To express the given expression in its lowest terms, we will follow these steps:

### Step 1: Factor the numerator and denominator
The expression we need to simplify is:

\[ \frac{(x-3)(x^2 - 5x + 4)}{(x-4)(x^2 - 2x - 3)} \]

First, we will factor \(x^2 - 5x + 4\) and \(x^2 - 2x - 3\).

**Factoring \(x^2 - 5x + 4\):**
We need to find two numbers that multiply to \(4\) (the constant term) and add to \(-5\) (the coefficient of \(x\)). The numbers \(-4\) and \(-1\) work:

\[ x^2 - 5x + 4 = (x - 4)(x - 1) \]

**Factoring \(x^2 - 2x - 3\):**
We need to find two numbers that multiply to \(-3\) and add to \(-2\). The numbers \(-3\) and \(1\) work:

\[ x^2 - 2x - 3 = (x - 3)(x + 1) \]

### Step 2: Substitute the factored forms back into the expression
Now we can rewrite the original expression using the factored forms:

\[ \frac{(x-3)(x-4)(x-1)}{(x-4)(x-3)(x+1)} \]

### Step 3: Cancel the common factors
We can see that \((x-3)\) and \((x-4)\) appear in both the numerator and the denominator, so we can cancel them:

\[ \frac{(x-1)}{(x+1)} \]

### Final Answer
Thus, the expression in its lowest terms is:

\[ \frac{x-1}{x+1} \]