#### Answer

$x=-2$

#### Work Step by Step

Separate the square roots on two sides. Then square both sides and combine like terms. Put all the terms on one side and factor by finding the GCF and trial and error. Then solve for $x$ with the values found. Only one of the answers is a solution when plugged into the original equation.
$\sqrt {x+2}+\sqrt {3x+7}=1$
$\sqrt {x+2}=1-\sqrt {3x+7}$
$(\sqrt {x+2})^2=(1-\sqrt {3x-7})^2$
$x+2=1-2\sqrt {3x+7}+3x+7$
$-2x-6=-2\sqrt {3x+7}$
$(-2x-6)^2=(-2\sqrt {3x+7})^2$
$4x^2+24x+36=4(3x+7)$
$4x^2+12x+8=0$
$4(x^2+3x+2)=0$
$4(x+2)(x+1)=0$
$x=-2,-1$
$\sqrt {-1+2}+\sqrt {3(-1)+7}\ne1$
$x=-2$