Question

How do I factor this expression completely?

`4x^2(x^2+1)(x+2)^3+36(x^2+1)^2(x+2)^4`

Answer 1

`4x^2(x^2+1)(x+2)^2+36(x^2+1)^2(x+2)^4=4(x^2+1)(x+2)^2(9x^4+36x^3+46x^2+36x+36)`

Explanation:

It is observed that `4x^2(x^2+1)(x+2)^2` and `36(x^2+1)^2(x+2)^4` have common factors

`4`, `(x^2+1)` and `(x+2)^2`.

Note that for `(x^2+1)` and `(x+2)`, we have selected the minimal power of these among two expressions. Similarly between `4` and `36`, `4` is common factor.

Hence taking out these common factors, we get

`4x^2(x^2+1)(x+2)^2+36(x^2+1)^2(x+2)^4`

= `4(x^2+1)(x+2)^2[x^2+9(x^2+1)(x+2)^2]`

= `4(x^2+1)(x+2)^2[x^2+9(x^2+1)(x^2+4x+4)]`

= `4(x^2+1)(x+2)^2[x^2+9(x^4+4x^3+4x^2+x^2+4x+4)]`

= `4(x^2+1)(x+2)^2(9x^4+36x^3+46x^2+36x+36)`