Answer:
[tex]\sqrt{x^2+2}\left(x^2+x+2\right)^2[/tex]
Step-by-step explanation:
Given equation:
[tex](x^2+2)^{\frac{5}{2}}+2x(x^2+2)^{\frac{3}{2}}+x^2\sqrt{x^2+2}[/tex]
[tex]\textsf{Rewrite the exponents }\frac{5}{2} \textsf{ as } \left(\frac{1}{2} \cdot 5\right)\textsf{ and }\frac{3}{2} \textsf{ as }\left(\frac{1}{2} \cdot 3\right):[/tex]
[tex]\implies (x^2+2)^{\frac{1}{2} \cdot 5}+2x(x^2+2)^{\frac{1}{2}\cdot 3}+x^2\sqrt{x^2+2}[/tex]
[tex]\textsf{Apply exponent rule} \quad a^{bc}=(a^b)^c:[/tex]
[tex]\implies \left((x^2+2)^{\frac{1}{2}}\right)^5+2x\left((x^2+2)^{\frac{1}{2}}\right)^3+x^2\sqrt{x^2+2}[/tex]
[tex]\textsf{Apply exponent rule} \quad a^{\frac{1}{2}}=\sqrt{a}:[/tex]
[tex]\implies \left(\sqrt{x^2+2}\right)^5+2x\left(\sqrt{x^2+2}\right)^3+x^2\sqrt{x^2+2}[/tex]
Factor out [tex]\sqrt{x^2+2}[/tex] from each of the three terms:
[tex]\implies \sqrt{x^2+2}\left[\left(\sqrt{x^2+2}\right)^4+2x\left(\sqrt{x^2+2}\right)^2+x^2\right][/tex]
[tex]\textsf{Factor the expression in the parentheses using} \quad a^2+2ab+b^2=(a+b)^2.[/tex]
[tex]\textsf{Let }a=\left(\sqrt{x^2+2}\right)^2 \textsf {and }b=x:[/tex]
[tex]\implies \sqrt{x^2+2}\left(\left(\sqrt{x^2+2}\right)^2+x\right)^2[/tex]
[tex]\textsf{Apply exponent rule} \quad \sqrt{a^2}=a:[/tex]
[tex]\implies \sqrt{x^2+2}\left(x^2+2+x\right)^2[/tex]
[tex]\implies \sqrt{x^2+2}\left(x^2+x+2\right)^2[/tex]
