Answer
$(x^{2}+4)^{3/2}(x^{4}+8x^{2}+17)$
Work Step by Step
$\left[(x^{2}+4)^{7/2}=(x^{2}+4)^{2+3/2}=(x^{2}+4)^{2}(x^{2}+4)^{3/2}\right]$
$(x^{2}+4)^{\frac{3}{2}}+(x^{2}+4)^{\frac{7}{2}}=(x^{2}+4)^{3/2}+(x^{2}+4)^{2}(x^{2}+4)^{3/2}$
... factor out $(x^{2}+4)^{3/2}$
$=(x^{2}+4)^{3/2}[1+(x^{2}+4)^{2}]$
... apply $(A+B)^{2}=A^{2}+2AB+B^{2}$
$=(x^{2}+4)^{3/2}(1+x^{4}+8x^{2}+16)$
$=(x^{2}+4)^{3/2}(x^{4}+8x^{2}+17)$
