Answer
$(x^{2}+4)^{3/2}(x^{4}+8x^{2}+17)$
Work Step by Step
$(x^{2}+4)^{3/2}+(x^{2}+4)^{7/2}$
The greatest common factor of $(x^{2}+4)^{3/2}+(x^{2}+4)^{7/2}$ is $(x^{2}+4)$ with the smaller exponent in the two terms. Thus the greatest common factor is $(x^{2}+4)^{3/2}$
Express each term as the product of greatest common factor and its other factor.
$=(x^{2}+4)^{3/2}+(x^{2}+4)^{3/2}(x^{2}+4)^{-3/2}(x^{2}+4)^{7/2}$
$[ (x^{2}+4)^{3/2}. (x^{2}+4)^{-3/2} = (x^{2}+4)^{3/2-3/2} = (x^{2}+4)^{0} = 1]$ because $[a^{m} . a^{n} = a^{m+n}]$
$=(x^{2}+4)^{3/2}+(x^{2}+4)^{3/2}(x^{2}+4)^{-3/2+7/2}$
$=(x^{2}+4)^{3/2}+(x^{2}+4)^{3/2}(x^{2}+4)^{2}$
Factor out the Greatest common factor.
$=(x^{2}+4)^{3/2}(1+(x^{2}+4)^{2})$
The square of the binomial can be found using the formula,
$(a+b)^{2}=a^{2}+2ab+b^{2}$
$=(x^{2}+4)^{3/2}(1+(x^{2})^{2}+2(x^{2})(4)+4^{2})$
$=(x^{2}+4)^{3/2}(1+x^{4}+8x^{2}+16)$
$=(x^{2}+4)^{3/2}(x^{4}+8x^{2}+17)$
