Rewrite the complex fraction as a division of rational expressions

$\frac{(\frac{1}{x+3}-\frac{1}{x})}{(1+\frac{3}{x})}$

$\frac{(\frac{1}{x+3}-\frac{1}{x})}{(1+\frac{3}{x})}=(\frac{1}{x+3}-\frac{1}{x})÷(1+\frac{3}{x})$

Simplify the numerator and denominators first

Numerator

$\frac{1}{x+3}\left(\frac{x}{x}\right)-\frac{1}{x}\left(\frac{x+3}{x+3}\right)$

$(\frac{x}{x(x+3)}-\frac{x+3}{x(x+3)})$

$\frac{x-x-3}{x(x+3)}$

$\frac{-3}{x(x+3)}$

Denominator

$(1\left(\frac{x}{x}\right)+\frac{3}{x})$

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

$\left(\frac{x+3}{x}\right)$

Rewrite

$\frac{-3}{x(x+3)}$ ÷ $\frac{x+3}{x}$

Find the reciprocal of the second fraction and multiply

$\frac{-3}{x(x+3)}\cdot \frac{x}{x+3}$

$\frac{-3x}{x(x+3)(x+3)}$

Cancel out like terms and simplify

$\frac{-3}{(x+3)(x+3)}$

$\frac{-3}{{(x+3)}^2}$

Multiply the numerator and denominator by a LCM of the numerator

$\frac{\frac{1}{x+3}-\frac{1}{x}}{1+\frac{3}{x}}$

The LCM of the numerator is x(x + 3)

$\frac{(\frac{1}{x+3}-\frac{1}{x})\cdot x(x+3)}{(1+\frac{3}{x})\cdot x(x+3)}$

Multiply and simplify by canceling out like terms

$\frac{\frac{1}{\cancel{x+3}}\cdot \frac{x(x+3)}{1}-\frac{1}{x}\cdot \frac{x(x+3)}{1}}{1\cdot x(x+3)+\frac{3}{x}\cdot x(x+3)}$

$\frac{x-(x+3)}{x(x+3)+3(x+3)}$

Simplify

$\frac{x-x-3}{(x+3)(x+3)}$

$\frac{-3}{{(x+3)}^2}$