\( \frac{42(3 x+5)^{2}(x-2)^{2}}{24(x-2)(3 x+5)^{3}} \) Puede dejar el numerador y denominador de la respuesta en forma factorizada.

Okay, I need to simplify the given expression and factorize both the numerator and the denominator. The expression is: \[ \frac{42(3x + 5)^2 (x - 2)^2}{24(x - 2)(3x + 5)^3} \] First, I'll look at the coefficients in the numerator and the denominator. The numerator has a coefficient of 42, and the denominator has a coefficient of 24. I can simplify these by dividing both by their greatest common divisor, which is 6. So, dividing numerator and denominator by 6: \[ \frac{42 \div 6}{24 \div 6} = \frac{7}{4} \] Now, the expression simplifies to: \[ \frac{7}{4} \cdot \frac{(3x + 5)^2 (x - 2)^2}{(x - 2)(3x + 5)^3} \] Next, I'll focus on the terms involving \( (3x + 5) \) and \( (x - 2) \). In the numerator, \( (3x + 5)^2 \) and in the denominator, \( (3x + 5)^3 \). I can simplify this by subtracting the exponents: \[ \frac{(3x + 5)^2}{(3x + 5)^3} = \frac{1}{3x + 5} \] Similarly, for \( (x - 2)^2 \) in the numerator and \( (x - 2) \) in the denominator: \[ \frac{(x - 2)^2}{(x - 2)} = x - 2 \] Putting it all together, the expression becomes: \[ \frac{7}{4} \cdot \frac{1}{3x + 5} \cdot (x - 2) = \frac{7(x - 2)}{4(3x + 5)} \] So, the simplified and factorized form of the original expression is: \[ \frac{7(x - 2)}{4(3x + 5)} \]