Factoring and Simplifying the Expression (m – 1)(m2 + m + 1)(m + 4)(m – 4)
This expression involves several factors, and we can simplify it by applying the concepts of difference of squares and sum of cubes.
Understanding the Factors
 (m  1): This is a simple linear factor.
 (m² + m + 1): This factor resembles the expansion of a cube: (a + b)³ = a³ + 3a²b + 3ab² + b³. However, it's not a perfect cube.
 (m + 4) and (m  4): These factors represent the difference of squares: a²  b² = (a + b)(a  b).
Applying the Concepts

Difference of Squares: Let's start by simplifying the last two factors: (m + 4)(m  4) = m²  16

Sum of Cubes (for a specific case): While (m² + m + 1) is not a perfect cube, we can use a specific case of the sum of cubes formula: a³ + b³ = (a + b)(a²  ab + b²)
Notice that if we set a = m and b = 1, we get: m³ + 1³ = (m + 1)(m²  m + 1)
Since we have (m²  m + 1), we can rewrite this as: m³ + 1 = (m + 1)(m²  m + 1)

Combining the Results: Now we can combine all the simplified factors: (m – 1)(m² + m + 1)(m + 4)(m – 4) = (m  1)(m²  m + 1)(m²  16)
Further Simplification
We can expand the last two factors, but it's not strictly necessary for simplification. The expression is now in a more compact and manageable form.
Final Simplified Form: (m  1)(m²  m + 1)(m²  16)
Important Note: This expression cannot be further factored into simpler linear factors using standard techniques.