The problem asks to find the Least Common Denominator (LCD) of two fractions: $\frac{14xy}{3x^2+4x}$ and $\frac{18y}{24x^2+29x-4}$.

1. Problem Description

The problem asks to find the Least Common Denominator (LCD) of two fractions: $\frac{14xy}{3x^2+4x}$ and $\frac{18y}{24x^2+29x-4}$.

2. Solution Steps

To find the LCD, we first need to factor the denominators.

First denominator:

$3x^2+4x = x(3x+4)$

Second denominator:

$24x^2+29x-4$. We look for two numbers that multiply to $(24)(-4) = -96$ and add up to $29$. Those numbers are $32$ and $-3$.

$24x^2+29x-4 = 24x^2 + 32x - 3x - 4 = 8x(3x+4) - 1(3x+4) = (8x-1)(3x+4)$

The factored denominators are $x(3x+4)$ and $(8x-1)(3x+4)$.

The LCD is the product of the unique factors, each raised to the highest power it appears in any of the denominators. Therefore, the LCD is $x(3x+4)(8x-1)$.

3. Final Answer

The LCD is $x(3x+4)(8x-1)$.