%5E2%2B3x(x%5E2%2B4x%2B8)%2B2x%5E2.jpeg "(x^2+4x+8)^2+3x(x^2+4x+8)+2x^2")
Simplifying the Expression: (x^2 + 4x + 8)^2 + 3x(x^2 + 4x + 8) + 2x^2
This article will guide you through the process of simplifying the given expression: (x^2 + 4x + 8)^2 + 3x(x^2 + 4x + 8) + 2x^2.
Step 1: Recognizing a Pattern
Observe that the expression contains a repeated term: (x^2 + 4x + 8). This pattern suggests a potential substitution to simplify the expression.
Step 2: Substitution
Let's introduce a new variable, y, to represent (x^2 + 4x + 8).
The expression now becomes: y^2 + 3xy + 2x^2.
Step 3: Factoring
The expression is now a quadratic in terms of y. We can factor it:
(y + 2x)(y + x)
Step 4: Back Substitution
Substitute (x^2 + 4x + 8) back in for y:
((x^2 + 4x + 8) + 2x)((x^2 + 4x + 8) + x)
Step 5: Simplifying
Expand the expression and combine like terms:
(x^2 + 6x + 8)(x^2 + 5x + 8)
Final Result
The simplified form of the given expression is (x^2 + 6x + 8)(x^2 + 5x + 8).