How do I factor this expression completely?

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

1 Answer
Jun 30, 2016

Answer:

#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)#