Simplifying the Expression (a+2)(a2)(a^2+2a+4)(a^22a+4)
This expression involves a pattern that allows us to simplify it significantly. Let's break it down:
Recognizing the Pattern
The expression consists of four factors:
 (a+2)(a2): This is a difference of squares pattern (a²  b²).
 (a² + 2a + 4): This is a sum of cubes pattern (a³ + b³).
 (a²  2a + 4): This is a difference of cubes pattern (a³  b³).
Applying the Patterns
Let's apply the algebraic patterns to simplify:

Difference of Squares: (a + 2)(a  2) = a²  2² = a²  4

Sum of Cubes: (a² + 2a + 4) = (a + 2)(a²  2a + 4)

Difference of Cubes: (a²  2a + 4) = (a  2)(a² + 2a + 4)
Combining the Results
Now we can substitute the simplified expressions back into the original one:
(a+2)(a2)(a^2+2a+4)(a^22a+4) = (a²  4) * (a + 2)(a²  2a + 4) * (a  2)(a² + 2a + 4)
Notice that the factors (a + 2) and (a  2) appear twice. Let's group them:
= (a²  4) * [(a + 2)(a  2)] * [(a²  2a + 4)(a² + 2a + 4)]
= (a²  4) * (a²  4) * (a⁴ + 4a² + 16)
Final Simplified Expression
Finally, we can multiply the remaining factors:
= (a⁴  8a² + 16) * (a⁴ + 4a² + 16)
This is the simplified form of the original expression. While it might seem complex, it's significantly easier to work with than the initial form, especially when performing further operations.