least common denominator of \( \frac{3}{x^{2}-8 x-33} \) and \( \frac{-2}{x^{2}-4} \)

To find the least common denominator (LCD) of the fractions \( \frac{3}{x^{2}-8 x-33} \) and \( \frac{-2}{x^{2}-4} \), we first factor the denominators. 1. For \( x^{2} - 8x - 33 \): We can factor this quadratic expression by finding two numbers that multiply to \(-33\) and add to \(-8\). These numbers are \(-11\) and \(3\). Therefore, \[ x^{2} - 8x - 33 = (x - 11)(x + 3). \] 2. For \( x^{2} - 4 \): This is a difference of squares, which can be factored as: \[ x^{2} - 4 = (x - 2)(x + 2). \] Now, we can express the denominators in factored form: - First fraction: \( (x - 11)(x + 3) \) - Second fraction: \( (x - 2)(x + 2) \) To find the least common denominator, we take each distinct factor with the highest exponent found in the factorizations: \[ \text{LCD} = (x - 11)(x + 3)(x - 2)(x + 2). \] Thus, the least common denominator of the two fractions is: \[ (x - 11)(x + 3)(x - 2)(x + 2). \]