1. **State the problem:** Simplify the expression $x^4 + 4x^2 - x^3y - 4xy$ by factoring. 2. **Identify common factors and group terms:** Group the terms as $(x^4 - x^3y) + (4x^2 - 4xy)$. 3. **Factor each group:** $$x^3(x - y) + 4x(x - y)$$ 4. **Factor out the common binomial factor:** $$\cancel{(x - y)}(x^3 + 4x)$$ 5. **Simplify the remaining factor:** $$x^3 + 4x = x(x^2 + 4)$$ 6. **Write the fully factored form:** $$x(x - y)(x^2 + 4)$$ 7. **Explanation:** We first grouped terms to find common factors, then factored each group, and finally factored out the common binomial $(x - y)$. The remaining factor $x^3 + 4x$ was factored by taking out $x$. The term $x^2 + 4$ cannot be factored further over the real numbers. **Final answer:** $$x(x - y)(x^2 + 4)$$