Question: Factor the expression $ 9x^2 - 30xy + 25y^2 - 16z^2 $ completely.

["Title: Complete Factorization of $ 9x^2 - 30xy + 25y^2 - 16z^2 $: A Step-by-Step Guide", "---", "Introduction", "Factoring algebraic expressions can sometimes feel challenging, especially when variables and coefficients multiply complexly. However, with a strategic approach, expressions like $ 9x^2 - 30xy + 25y^2 - 16z^2 $ become manageable and even elegant. In this article, we’ll explore how to fully factor the expression $ 9x^2 - 30xy + 25y^2 - 16z^2 $, breaking it down step by step using techniques ideal for recognizing perfect squares and difference of squares.", "---", "### Step 1: Group Terms for Pattern Recognition", "The expression is:", "$$\n9x^2 - 30xy + 25y^2 - 16z^2\n$$", "Notice the first three terms—$ 9x^2 - 30xy + 25y^2 $—form a perfect square trinomial. Check if:", "$$\n9x^2 - 30xy + 25y^2 = (3x - 5y)^2\n$$", "Indeed:", "$$\n(3x - 5y)^2 = 9x^2 - 2(3x)(5y) + (5y)^2 = 9x^2 - 30xy + 25y^2\n$$", "So, we rewrite the original expression as:", "$$\n(3x - 5y)^2 - 16z^2\n$$", "---", "### Step 2: Recognize Differencing of Squares", "Now the expression is in the form:", "$$\na^2 - b^2\n$$", "where:", "- $ a = (3x - 5y) $\n- $ b = 4z $, since $ 16z^2 = (4z)^2 $", "Recall the identity:", "$$\na^2 - b^2 = (a - b)(a + b)\n$$", "Apply this:", "$$\n(3x - 5y)^2 - (4z)^2 = \left( (3x - 5y) - 4z \right)\left( (3x - 5y) + 4z \right)\n$$", "---", "### Step 3: Write Final Factored Form", "Thus, the complete factorization is:", "$$\n9x^2 - 30xy + 25y^2 - 16z^2 = (3x - 5y - 4z)(3x - 5y + 4z)\n$$", "---", "### Why This Approach Works", "1. Grouping by Perfect Squares: Identifying $ 9x^2 - 30xy + 25y^2 $ as $ (3x - 5y)^2 $ simplifies the expression significantly.\n2. Difference of Squares Identification: Recognizing the final expression as $ a^2 - b^2 $ allows direct application of the powerful identity, avoiding more complex factoring attempts.\n3. Clear, Structured Steps: Each step builds logically from the previous one, making the factorization easy to follow and verify.", "---", "### Key Takeaways", "- Always look for perfect squares during factoring— they often hide behind apparent quadratic forms.\n- Grouping terms efficiently reveals familiar algebraic patterns.\n- The difference of squares remains one of the most reliable methods for complete factorization.", "---", "Conclusion", "Factoring $ 9x^2 - 30xy + 25y^2 - 16z^2 $ reduces elegantly to recognizing a perfect square minus another square, which then applies the identity $ a^2 - b^2 = (a - b)(a + b) $. The fully factored expression is:", "$$\n\boxed{(3x - 5y - 4z)(3x - 5y + 4z)}\n$$", "Mastering such techniques strengthens algebraic intuition and enables confident handling of similarly structured expressions.", "---", "Keywords for SEO:\nfactor $ 9x^2 - 30xy + 25y^2 - 16z^2 $, factor completely, perfect square trinomial, difference of squares, algebraic factoring, step-by-step factorization, $ (3x - 5y - 4z)(3x - 5y + 4z) $, complete factorization, identify perfect square, apply difference of squares.", "---", "OngoVitally — Your go-to resource for clear, precise, and optimized algebra explanations."]