$$ m\left(x+y\right)^2-4m\left(x-y\right)^2 $$

We are given the expression:

\[
m\left(x+y\right)^2 - 4m\left(x-y\right)^2
\]

Let's simplify this step by step:

### Step 1: Distribute the $ m $

First, factor $ m $ from both terms:

\[
m \left( \left(x + y\right)^2 - 4\left(x - y\right)^2 \right)
\]

### Step 2: Expand the squares

Expand both $\left(x + y\right)^2$ and $\left(x - y\right)^2$:

\[
\left(x + y\right)^2 = x^2 + 2xy + y^2
\]
\[
\left(x - y\right)^2 = x^2 - 2xy + y^2
\]

### Step 3: Substitute these into the expression

Now substitute these into the original equation:

\[
m \left( (x^2 + 2xy + y^2) - 4(x^2 - 2xy + y^2) \right)
\]

### Step 4: Distribute the -4

Distribute the -4 to each term in the second square:

\[
m \left( (x^2 + 2xy + y^2) - (4x^2 - 8xy + 4y^2) \right)
\]

### Step 5: Combine like terms

Now simplify by combining like terms:

\[
m \left( x^2 + 2xy + y^2 - 4x^2 + 8xy - 4y^2 \right)
\]

\[
m \left( (x^2 - 4x^2) + (2xy + 8xy) + (y^2 - 4y^2) \right)
\]

This simplifies to:

\[
m \left( -3x^2 + 10xy - 3y^2 \right)
\]

### Final answer:

The simplified expression is:

\[
m \left( -3x^2 + 10xy - 3y^2 \right)
\]

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you expand binomials like $(a+b)^2$?
2. How do you factor quadratic expressions?
3. Can you derive similar expressions with higher powers, like $(x+y)^3$?
4. What is the role of the distributive property in algebra?
5. How do you solve quadratic equations?

**Tip:** Always simplify step by step to avoid mistakes when expanding or factoring expressions!

Explore how to simplify the algebraic expression m(x+y)^2 - 4m(x-y)^2. This step-by-step solution covers key algebraic concepts, including binomial expansion and the distributive property. Suitable for students in Grades 9-12.

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