**64y6** **- 112y3z7** **+ 49z1436f2** **- 64g2**

The expression you've provided is:

\[ 64y^6 - 112y^3z^7 + 49z^{14}36f^2 - 64g^2 \]

This expression has four terms and looks like a candidate for factoring by grouping or recognizing specific patterns.

### Step 1: Group the terms
First, group the terms in pairs:
\[ (64y^6 - 112y^3z^7) + (49z^{14}36f^2 - 64g^2) \]

### Step 2: Factor each group
Let's factor each group individually.

#### Group 1: \( 64y^6 - 112y^3z^7 \)
Factor out the common factor \( 16y^3 \):
\[ 16y^3(4y^3 - 7z^7) \]

#### Group 2: \( 49z^{14}36f^2 - 64g^2 \)
Notice that this can be rewritten as:
\[ (7z^7 \times 6f)^2 - (8g)^2 \]

This is a difference of squares, which factors as:
\[ (7z^7 \times 6f + 8g)(7z^7 \times 6f - 8g) \]

### Step 3: Combine the factors
The fully factored expression is:
\[ 16y^3(4y^3 - 7z^7) + (7z^7 \times 6f + 8g)(7z^7 \times 6f - 8g) \]

This expression has been simplified as far as it can be factored directly.

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Would you like more details or have any questions?

Here are 8 related questions you might find interesting:
1. What is the difference of squares, and how do you identify it?
2. How does factoring by grouping work?
3. What are common factoring techniques for polynomials?
4. Can you factor more complex polynomials with more than four terms?
5. How do you determine which terms to group together?
6. What is the next step if factoring by grouping doesn't work?
7. Can every polynomial be factored over the integers?
8. How can you verify your factored expression is correct?

**Tip:** Always check if your expression can be factored by looking for common factors, differences of squares, and perfect square trinomials!

Learn how to factor the polynomial expression 64y^6 - 112y^3z^7 + 49z^1436f^2 - 64g^2 step-by-step. This problem involves factoring by grouping and recognizing the difference of squares pattern. Suitable for advanced high school students and beyond.

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