Question

Factorize the equation

Answer 1

`√(x^2+2)\left(x^2+x+2\right)^2`

Explanation:

Given equation:

`(x^2+2)^{(5)/(2)}+2x(x^2+2)^{(3)/(2)}+x^2√(x^2+2)`

`\textsf{Rewrite the exponents }(5)/(2) \textsf{ as } \left((1)/(2) \cdot 5\right)\textsf{ and }(3)/(2) \textsf{ as }\left((1)/(2) \cdot 3\right):`

`\implies (x^2+2)^{(1)/(2) \cdot 5}+2x(x^2+2)^{(1)/(2)\cdot 3}+x^2√(x^2+2)`

`\textsf{Apply exponent rule} \quad a^(bc)=(a^b)^c:`

`\implies \left((x^2+2)^{(1)/(2)}\right)^5+2x\left((x^2+2)^{(1)/(2)}\right)^3+x^2√(x^2+2)`

`\textsf{Apply exponent rule} \quad a^{(1)/(2)}=√(a):`

`\implies \left(√(x^2+2)\right)^5+2x\left(√(x^2+2)\right)^3+x^2√(x^2+2)`

Factor out
`√(x^2+2)` from each of the three terms:

`\implies √(x^2+2)\left[\left(√(x^2+2)\right)^4+2x\left(√(x^2+2)\right)^2+x^2\right]`

`\textsf{Factor the expression in the parentheses using} \quad a^2+2ab+b^2=(a+b)^2.`

`\textsf{Let }a=\left(√(x^2+2)\right)^2 \textsf {and }b=x:`

`\implies √(x^2+2)\left(\left(√(x^2+2)\right)^2+x\right)^2`

`\textsf{Apply exponent rule} \quad √(a^2)=a:`

`\implies √(x^2+2)\left(x^2+2+x\right)^2`

`\implies √(x^2+2)\left(x^2+x+2\right)^2`