(m^2-n^2m)^2+2m^3n^2

Jasonbradley

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Jun 16, 2024

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Factoring the Expression (m^2-n^2m)^2+2m^3n^2

This article explores the factorization of the expression (m^2-n^2m)^2+2m^3n^2. We'll break down the steps and reveal the factored form.

Understanding the Expression

The expression is a sum of two terms:

• (m^2-n^2m)^2: This is the square of a binomial.
• 2m^3n^2: This is a simple monomial.

Our goal is to rewrite the expression in a simpler form by factoring out common factors.

Step-by-Step Factorization

1. Recognize the Pattern: Notice that the first term, (m^2-n^2m)^2, is a perfect square trinomial. It fits the pattern (a-b)^2 = a^2 - 2ab + b^2.

2. Expand the Square: Let's expand the first term:

   (m^2-n^2m)^2 = (m^2)^2 - 2(m^2)(n^2m) + (n^2m)^2 = m^4 - 2m^3n^2 + m^2n^4

3. Rewrite the Expression: Now, substitute this expanded term back into the original expression:

   (m^2-n^2m)^2+2m^3n^2 = m^4 - 2m^3n^2 + m^2n^4 + 2m^3n^2

4. Simplify: Notice that the -2m^3n^2 and +2m^3n^2 terms cancel each other out:

   m^4 - 2m^3n^2 + m^2n^4 + 2m^3n^2 = m^4 + m^2n^4

5. Factor Out Common Factors: Both terms have a common factor of m^2:

   m^4 + m^2n^4 = m^2(m^2 + n^4)

The Factored Form

Therefore, the factored form of (m^2-n^2m)^2+2m^3n^2 is m^2(m^2 + n^4).