Let's simplify the expression \(16x^2y - 20xy - 50y\).

### Step 1: Factor out the greatest common factor (GCF)

First, identify the GCF of all the terms. The terms are \(16x^2y\), \(-20xy\), and \(-50y\).

- The GCF of the numerical coefficients \(16\), \(-20\), and \(-50\) is \(2\).
- The variable part common to all terms is \(y\).

Thus, the GCF of the entire expression is \(2y\).

### Step 2: Factor out the GCF

Factor \(2y\) from each term:

\[
16x^2y - 20xy - 50y = 2y(8x^2 - 10x - 25)
\]

### Step 3: Factor the quadratic expression, if possible

Now, let's try to factor \(8x^2 - 10x - 25\).

1. The expression is a quadratic trinomial in the form \(ax^2 + bx + c\), where \(a = 8\), \(b = -10\), and \(c = -25\).
2. To factor it, we look for two numbers that multiply to \(8 \times -25 = -200\) and add to \(-10\). The factors of \(-200\) that add up to \(-10\) are not easily identifiable, suggesting this expression does not factor nicely over the integers.

Since \(8x^2 - 10x - 25\) doesn't factor nicely, the simplified form of the expression remains:

\[
\boxed{2y(8x^2 - 10x - 25)}
\]

Would you like to see more details or have any questions? Here are some related questions you might explore:

1. How do you factor a quadratic trinomial when factoring directly isn't possible?
2. How can you apply the quadratic formula to solve equations of the form \(ax^2 + bx + c\)?
3. What methods can you use if a trinomial doesn’t factor easily?
4. How do you identify and factor the greatest common factor in complex expressions?
5. What are some tips for checking if your factorization is correct?

**Tip:** Always start with factoring out the GCF before attempting to factor more complex parts of an expression.

Learn how to factor the expression 16x^2y - 20xy - 50y by identifying the greatest common factor (GCF) and simplifying the quadratic expression. This step-by-step solution covers key algebraic concepts, including factoring and quadratic expressions. Suitable for students in Grades 9-10.

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