Let's break this problem down step by step.

Given expression:

$10n^3 - 15n^2 + 20xn^2 - 30xn$

Part A: Factoring Out the Greatest Common Factor (GCF)

1. Identify the GCF:

   • Look at the coefficients: 10, 15, 20, and 30. The greatest common factor of these numbers is 5.
   • Look at the variables: Each term contains at least one factor of n, so n is also part of the GCF.

   Therefore, the GCF of the entire expression is 5n.

2. Factor out the GCF: To factor out $5n$, divide each term of the expression by $5n$:

$10n^3 ÷ 5n = 2n^2$ $15n^2 ÷ 5n = 3n$ $20xn^2 ÷ 5n = 4xn$ $30xn ÷ 5n = 6x$

After factoring out $5n$, the expression becomes:

$5n(2n^2 - 3n + 4xn - 6x)$

Part B: Factoring the Entire Expression Completely

1. Rewrite the expression after factoring out the GCF: We now need to factor the quadratic-like expression inside the parentheses:

$2n^2 - 3n + 4xn - 6x$

2. Group the terms: To factor this completely, we will try grouping the terms: $(2n^2 - 3n) + (4xn - 6x)$

3. Factor out the GCF from each group:

   • From $2n^2 - 3n$, the GCF is $n$, so we get $n(2n - 3)$.
   • From $4xn - 6x$, the GCF is $2x$, so we get $2x(2n - 3)$.

   Now, the expression becomes:

   $n(2n - 3) + 2x(2n - 3)$

4. Factor by grouping: Notice that both terms now have a common factor of $(2n - 3)$. We can factor this out:

   $(2n - 3)(n + 2x)$

Thus, the fully factored expression is:

$5n(2n - 3)(n + 2x)$

Final Answer:

• Part A: The expression factored by the GCF is: $5n(2n^2 - 3n + 4xn - 6x)$

• Part B: The expression completely factored is: $5n(2n - 3)(n + 2x)$

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Would you like more details on any specific part of this factorization?

Here are 5 related questions to explore this further:

1. How do you determine the GCF of a polynomial with both coefficients and variables?
2. Can all quadratic expressions be factored by grouping, or are there exceptions?
3. How would the factoring process change if the polynomial had a different number of terms?
4. What are the signs to watch for when deciding if an expression can be factored by grouping?
5. How can factoring help solve equations or inequalities involving polynomials?

Tip: When factoring by grouping, always check for a common factor across all terms before starting the grouping process!