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To factor the quadratic expression \(x^2 - 2x - 63\), we need to find two numbers that multiply to \(-63\) and add up to \(-2\).<br /><br />Let's analyze the given options:<br /><br />1. \((x-9)(x+2)\)<br /> - Multiplying: \((x-9)(x+2) = x^2 + 2x - 9x - 18 = x^2 - 7x - 18\)<br /> - This does not match \(x^2 - 2x - 63\).<br /><br />2. \((x+9)(x-7)\)<br /> - Multiplying: \((x+9)(x-7) = x^2 - 7x + 9x - 63 = x^2 + 2x - 63\)<br /> - This does not match \(x^2 - 2x - 63\).<br /><br />3. \((x-2)(x+9)\)<br /> - Multiplying: \((x-2)(x+9) = x^2 + 9x - 2x - 18 = x^2 + 7x - 18\)<br /> - This does not match \(x^2 - 2x - 63\).<br /><br />4. \((x-9)(x+7)\)<br /> - Multiplying: \((x-9)(x+7) = x^2 + 7x - 9x - 63 = x^2 - 2x - 63\)<br /> - This matches \(x^2 - 2x - 63\).<br /><br />Therefore, the correct factorization is \((x-9)(x+7)\).