(x^2+4x+8)^2 +3x( x^2 +4x+8) + 2x^2

(x + 2) (x + 4) (x^2 + 5 x + 8)

Step-by-step explanation:

Simplify the following:

(x^2 + 4 x + 8)^2 + 3 x (x^2 + 4 x + 8) + 2 x^2

(x^2 + 4 x + 8)^2 + 3 x (x^2 + 4 x + 8) + 2 x^2 = 2 x^2 + 3 x (x^2 + 4 x + 8) + 1 (x^2 + 4 x + 8)^2:

2 x^2 + 3 x (x^2 + 4 x + 8) + (x^2 + 4 x + 8)^2

The factors of 2 that sum to 3 are 2 and 1. So, 2 x^2 + 3 x (x^2 + 4 x + 8) + (x^2 + 4 x + 8)^2 = ((x^2 + 4 x + 8) + 2 x) ((x^2 + 4 x + 8) + 1 x):

(x^2 + 4 x + 2 x + 8) (x^2 + 4 x + x + 8)

Grouping like terms, x^2 + 4 x + x + 8 = x^2 + (4 x + x) + 8:

x^2 + (4 x + x) + 8 (x^2 + 4 x + 2 x + 8)

4 x + x = 5 x:

(x^2 + 5 x + 8) (x^2 + 4 x + 2 x + 8)

Grouping like terms, x^2 + 4 x + 2 x + 8 = x^2 + (4 x + 2 x) + 8:

x^2 + (4 x + 2 x) + 8 (x^2 + 5 x + 8)

4 x + 2 x = 6 x:

(x^2 + 6 x + 8) (x^2 + 5 x + 8)

The factors of 8 that sum to 6 are 4 and 2. So, x^2 + 6 x + 8 = (x + 4) (x + 2):

(x + 2) (x + 4) (x^2 + 5 x + 8)