The given expression is:

\[
55xy^2 + 110x^2y - 220x^2y^2m
\]

We can factor this expression by first looking for the greatest common factor (GCF) of all the terms.

### Step 1: Identify the GCF
The terms have numerical coefficients of 55, 110, and 220. The GCF of these numbers is 55. We also notice that \(x\) and \(y\) appear in all the terms, so we can factor out \(x\) and \(y\). The GCF in terms of the variables would be \(x\) and \(y\), considering the lowest powers of each variable.

So, the GCF of the expression is \(55xy\).

### Step 2: Factor out the GCF
Now, we factor \(55xy\) from each term:

\[
55xy^2 + 110x^2y - 220x^2y^2m = 55xy \left( y + 2x - 4xym \right)
\]

### Final factored form:

\[
55xy \left( y + 2x - 4xym \right)
\]

This is the factored version of the given expression.

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Would you like more details on factoring or simplification? 

Here are 5 related questions that might help you expand on this topic:
1. How do you factor polynomials with higher powers of variables?
2. What is the process for finding the greatest common factor (GCF) in more complex expressions?
3. How can you apply factoring in solving algebraic equations?
4. How does factoring simplify expressions in calculus?
5. How would this expression behave if specific values were assigned to \(x\), \(y\), and \(m\)?

**Tip**: Always double-check your factorization by multiplying the terms back to ensure the original expression is recovered!

Learn how to factor the polynomial 55xy^2 + 110x^2y - 220x^2y^2m step by step. This algebra problem focuses on finding the greatest common factor (GCF) and using the distributive property. Suitable for students in Grades 9-11.

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