Step-by-Step Solution
Key Concept: A fraction equals zero if and only if its numerator is zero AND its denominator is non-zero. Factor both polynomials completely, then exclude any solutions that make the denominator zero.
<p><strong>Step 1:</strong> Factor the numerator: x² + 3x + 2 = (x + 1)(x + 2)</p><p><strong>Step 2:</strong> Factor the denominator: x² - 6x - 7 = (x - 7)(x + 1)</p><p><strong>Step 3:</strong> The equation becomes: $\frac{(x+1)(x+2)}{(x-7)(x+1)} = 0$</p><p><strong>Step 4:</strong> For a fraction to equal zero, numerator = 0 and denominator ≠ 0. The numerator is zero when x = -1 or x = -2.</p><p><strong>Step 5:</strong> Check denominators: When x = -1, denominator = (-1 - 7)(-1 + 1) = 0 ✗ (extraneous). When x = -2, denominator = (-2 - 7)(-2 + 1) = (-9)(-1) = 9 ≠ 0 ✓</p><p>∴ Answer: x = -2</p>
Correct Answer: x = -2