The problem asks us to express the expression $\frac{4}{x+5} - \frac{3}{x}$ as a single fraction in its lowest simplest form.

1. Problem Description

The problem asks us to express the expression $\frac{4}{x+5} - \frac{3}{x}$ as a single fraction in its lowest simplest form.

2. Solution Steps

First, we find a common denominator for the two fractions, which is $x(x+5)$.

Then, we rewrite each fraction with the common denominator:

$\frac{4}{x+5} = \frac{4x}{x(x+5)}$

$\frac{3}{x} = \frac{3(x+5)}{x(x+5)}$

Now, we can subtract the two fractions:

$\frac{4}{x+5} - \frac{3}{x} = \frac{4x}{x(x+5)} - \frac{3(x+5)}{x(x+5)} = \frac{4x - 3(x+5)}{x(x+5)}$

Next, we simplify the numerator:

$4x - 3(x+5) = 4x - 3x - 15 = x - 15$

So, the expression becomes:

$\frac{x-15}{x(x+5)}$

Finally, we expand the denominator:

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

The simplified fraction is:

$\frac{x-15}{x^2+5x}$

3. Final Answer

$\frac{x-15}{x^2+5x}$