Factor completely.
6 x squared y minus 34 xy squared plus 40 y cubed
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Part 1
6 x squared y minus 34 xy squared plus 40 y cubedequals
  
2 y left parenthesis x minus 4 y right parenthesis left parenthesis 3 x minus 5 y right parenthesis

To factor the expression \(6x^2y - 34xy^2 + 40y^3\) completely, let's go through it step-by-step.

### Step 1: Factor Out the Greatest Common Factor
The terms \(6x^2y\), \(-34xy^2\), and \(40y^3\) each have a common factor of \(2y\).
\[
6x^2y - 34xy^2 + 40y^3 = 2y(3x^2 - 17xy + 20y^2)
\]

### Step 2: Factor the Quadratic Trinomial
Now we focus on factoring the quadratic trinomial inside the parentheses: \(3x^2 - 17xy + 20y^2\).

We need to find two numbers that multiply to \((3 \times 20)y^2 = 60y^2\) and add to \(-17y\).
The numbers \(-12y\) and \(-5y\) satisfy these conditions:
\[
-12y \times -5y = 60y^2 \quad \text{and} \quad -12y + (-5y) = -17y
\]

So, we can split the middle term:
\[
3x^2 - 17xy + 20y^2 = 3x^2 - 12xy - 5xy + 20y^2
\]

### Step 3: Factor by Grouping
Now, group the terms:
\[
= (3x^2 - 12xy) - (5xy - 20y^2)
\]

Factor out the common factors from each group:
\[
= 3x(x - 4y) - 5y(x - 4y)
\]

Now, factor out \((x - 4y)\):
\[
= (x - 4y)(3x - 5y)
\]

### Step 4: Substitute Back
Now, substitute back into our original expression:
\[
6x^2y - 34xy^2 + 40y^3 = 2y(x - 4y)(3x - 5y)
\]

### Final Answer
The completely factored form is:
\[
\boxed{2y(x - 4y)(3x - 5y)}
\]

Would you like further details on any steps or have additional questions?

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Here are some related questions that might interest you:

1. How can you verify that a factorization is correct?
2. How do you handle factoring when variables are raised to higher powers?
3. What if the quadratic does not factor easily? Are there alternative methods?
4. How can factoring be used to simplify complex expressions in calculus?
5. How does factoring relate to solving quadratic equations?

**Tip:** Always start by factoring out the greatest common factor (GCF) from all terms, as it simplifies the remaining steps.

Factor completely. 6x^2y - 34xy^2 + 40y^3

Explore how to factor the expression 6x^2y - 34xy^2 + 40y^3 completely with a detailed step-by-step solution. This problem involves factoring by grouping, applying the greatest common factor, and breaking down trinomials, ideal for students in Grades 8-10.

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