Answer
$(-1)(x+2)(x+3)^{1/2}$
Work Step by Step
$(x+3)^{1/2}-(x+3)^{3/2}$
The greatest common factor of $(x+3)^{1/2}-(x+3)^{3/2}$ is $(x+3)$ with the smaller exponent in the two terms. Thus the greatest common factor is $(x+3)^{1/2}$
Express each term as the product of greatest common factor and its other factor.
$=(x+3)^{1/2}- (x+3)^{1/2} (x+3)^{-1/2}(x+3)^{3/2} $ $[(x+3)^{1/2} (x+3)^{-1/2} = (x+3)^{1/2 -1/2} = (x+3)^{0} = 1]$
$=(x+3)^{1/2}- (x+3)^{1/2}(x+3) $ $[(x+3)^{3/2 -1/2} = (x+3)^{2/2} = (x+3)^{1} ]$ because $[a^{m}. a^{n} = a^{m+n}]$
Factor out the Greatest common factor.
$= (x+3)^{1/2}(1-(x+3))$
$= (x+3)^{1/2}(1-x-3)$
$= (x+3)^{1/2}(-x-2)$
$= (x+3)^{1/2}(-1)(x+2)$
$= (-1)(x+2)(x+3)^{1/2}$
