The problem asks to find the Least Common Denominator (LCD) of the two fractions $\frac{18xy}{4x^2+5x}$ and $\frac{24y}{20x^2+17x-10}$. The answer should be in factored form.

1. Problem Description

The problem asks to find the Least Common Denominator (LCD) of the two fractions $\frac{18xy}{4x^2+5x}$ and $\frac{24y}{20x^2+17x-10}$. The answer should be in factored form.

2. Solution Steps

First, we need to factor the denominators of the two fractions.

The first denominator is $4x^2 + 5x$. We can factor out an $x$:

$4x^2 + 5x = x(4x + 5)$

The second denominator is $20x^2 + 17x - 10$. We are looking for two numbers that multiply to $20(-10)=-200$ and add up to $17$. The two numbers are $25$ and $-8$.

$20x^2 + 17x - 10 = 20x^2 + 25x - 8x - 10 = 5x(4x + 5) - 2(4x + 5) = (5x - 2)(4x + 5)$

So the two denominators in factored form are:

$x(4x + 5)$ and $(5x - 2)(4x + 5)$

The LCD is the product of the highest power of each unique factor present in the denominators. The factors are $x$, $(4x+5)$, and $(5x-2)$. Therefore, the LCD is $x(4x + 5)(5x - 2)$.

3. Final Answer

The LCD is $x(4x+5)(5x-2)$.