14\left(3x-5y\right)^3+7\left(3x-5y\right)^2

The expression you've provided is:

\[
14\left(3x - 5y\right)^3 + 7\left(3x - 5y\right)^2
\]

This expression can be factored by recognizing the common factor \(\left(3x - 5y\right)^2\). Here’s how to factor it:

1. Identify the common factor \(\left(3x - 5y\right)^2\) in both terms.

\[
14\left(3x - 5y\right)^3 + 7\left(3x - 5y\right)^2 = \left(3x - 5y\right)^2 \left(14\left(3x - 5y\right) + 7\right)
\]

2. Simplify the expression inside the parentheses:

\[
14\left(3x - 5y\right) + 7 = 42x - 70y + 7
\]

So the factored expression is:

\[
\left(3x - 5y\right)^2 \left(42x - 70y + 7\right)
\]

Would you like more details or have any questions?

Here are 8 related questions you might find interesting:
1. How would you expand the factored expression \(\left(3x - 5y\right)^2 \left(42x - 70y + 7\right)\)?
2. Can you factor the expression if the coefficients were different, say \(14(4x - 5y)^3 + 7(4x - 5y)^2\)?
3. What happens if the original expression has a negative sign between the two terms?
4. How does factoring help in solving algebraic equations?
5. Can this factoring technique be applied to quadratic expressions?
6. What if there were more terms in the original expression?
7. How would the factorization change if the exponents were different?
8. Can you apply this factorization method to trigonometric expressions?

**Tip:** When factoring expressions, always check for the greatest common factor (GCF) first. This can simplify the factorization process.

Understand how to factorize the expression 14(3x - 5y)^3 + 7(3x - 5y)^2 with detailed step-by-step instructions. This problem involves algebraic factorization techniques, focusing on identifying common factors and simplifying algebraic expressions. Suitable for students in Grades 9-12.

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