For all `x > 21, ((x^(2) + 8x + 7)(x - 3))/((x^(2) + 4x - 21)(x + 1)) = ?`

Options

A. 1
B. `9/7`
C. `(x-3)/(x+3)`
D. `(2(x-3))/(x+1)`

Answer

To simplify the given expression \(\frac{(x^2 + 8x + 7)(x - 3)}{(x^2 + 4x - 21)(x + 1)}\) for \(x > 21\), we will follow these steps:

### Step 1: Factor the Numerator
We start with the numerator \(x^2 + 8x + 7\).

1. **Identify the quadratic expression**: \(x^2 + 8x + 7\).
2. **Factor it**: We look for two numbers that multiply to \(7\) and add up to \(8\). The numbers \(1\) and \(7\) work.
\[ x^2 + 8x + 7 = (x + 1)(x + 7) \]

### Step 2: Rewrite the Numerator
Now, we can rewrite the numerator using the factors we found:
\[ (x^2 + 8x + 7)(x - 3) = (x + 1)(x + 7)(x - 3) \]

### Step 3: Factor the Denominator
Next, we factor the denominator \(x^2 + 4x - 21\).

1. **Identify the quadratic expression**: \(x^2 + 4x - 21\).
2. **Factor it**: We need two numbers that multiply to \(-21\) and add to \(4\). The numbers \(7\) and \(-3\) work.
\[ x^2 + 4x - 21 = (x + 7)(x - 3) \]

### Step 4: Rewrite the Denominator
Now, we can rewrite the denominator using the factors we found:
\[ (x^2 + 4x - 21)(x + 1) = (x + 7)(x - 3)(x + 1) \]

### Step 5: Substitute Back into the Expression
Now we substitute the factored forms back into the original expression:
\[ \frac{(x + 1)(x + 7)(x - 3)}{(x + 7)(x - 3)(x + 1)} \]

### Step 6: Cancel Common Factors
We can now cancel the common factors in the numerator and the denominator:
- The factors \((x + 1)\), \((x + 7)\), and \((x - 3)\) are present in both the numerator and the denominator.

After cancellation, we are left with:
\[ 1 \]

### Final Result
Thus, the simplified expression is:
\[ \frac{(x^2 + 8x + 7)(x - 3)}{(x^2 + 4x - 21)(x + 1)} = 1 \]