#### Answer

$\displaystyle x=-\frac{7}{3}$ or $x=-4$

#### Work Step by Step

$\displaystyle \frac{x}{2x+7}-\frac{x+1}{x+3}=1$
We multiply through by $(2x+7)(x+3)$:
$x(x+3)-(x+1)(2x+7)=(2x+7)(x+3)$
And distribute:
$x^{2}+3x-2x^{2}-9x-7=2x^{2}+13x+21$
And simplify:
$x^{2}+3x-2x^{2}-9x-7-2x^{2}-13x-21=0$
$3x^{2}+19x+28=0$
And factor:
$(3x+7)(x+4)=0$
$3x+7=0$ or $x+4=0$
$x=-\frac{7}{3}$ or $x=-4$