Certainly! Let's analyze and factor the given expression step by step.
Given expression:
[tex]\[ 10n^3 - 15n^2 + 20xn^2 - 30xn \][/tex]
### Part A: Rewrite the expression by factoring out the greatest common factor (GCF).
Step 1: Identify the constants/coefficients and variables involved.
- Coefficients: [tex]\(10, -15, 20, -30\)[/tex]
- Variables: [tex]\(n^3, n^2, xn^2, xn\)[/tex]
Step 2: Find the GCF of the coefficients.
- The GCF of [tex]\(10, 15, 20,\)[/tex] and [tex]\(30\)[/tex] is [tex]\(5\)[/tex].
Step 3: Find the GCF of the variable terms.
- All terms have at least one [tex]\(n\)[/tex]. The lowest power of [tex]\(n\)[/tex] among the terms is [tex]\(n\)[/tex].
Combining these, the GCF of the entire expression is [tex]\(5n\)[/tex].
Step 4: Factor out the GCF from each term.
[tex]\[
10n^3 - 15n^2 + 20xn^2 - 30xn = 5n(2n^2 - 3n + 4xn - 6x)
\][/tex]
Thus, the factored expression by removing the GCF is:
[tex]\[
5n(2n^2 - 3n + 4xn - 6x)
\][/tex]
### Part B: Factor the entire expression completely.
Step 5: Simplify inside the parentheses.
We need to check if we can factor further inside [tex]\(2n^2 - 3n + 4xn - 6x\)[/tex].
Rearrange the terms within the parentheses to group them for factoring:
[tex]\[
2n^2 - 3n + 4xn - 6x = 2n^2 + 4xn - 3n - 6x
\][/tex]
Step 6: Factor by grouping.
Group the terms in pairs:
[tex]\[
2n^2 + 4xn - 3n - 6x = (2n^2 + 4xn) + (-3n - 6x)
\][/tex]
Factor out common factors in each group:
[tex]\[
(2n^2 + 4xn) + (-3n - 6x) = 2n(n + 2x) - 3(n + 2x)
\][/tex]
Step 7: Notice [tex]\(n + 2x\)[/tex] is a common factor in the groups.
[tex]\[
2n(n + 2x) - 3(n + 2x) = (2n - 3)(n + 2x)
\][/tex]
Therefore, the fully factored expression inside the parentheses is:
[tex]\[
2n^2 - 3n + 4xn - 6x = (2n - 3)(n + 2x)
\][/tex]
Step 8: Substitute back into the original factored expression.
Thus, the completely factored form of the original expression [tex]\(10n^3 - 15n^2 + 20xn^2 - 30xn\)[/tex] is:
[tex]\[
5n(2n - 3)(n + 2x)
\][/tex]
### Final Answer:
Part A:
[tex]\[
5n(2n^2 - 3n + 4xn - 6x)
\][/tex]
Part B:
[tex]\[
5n(2n - 3)(n + 2x)
\][/tex]
