To rewrite the expression by factoring out the greatest common factor, we'll follow these steps:
### Step 1: Identify the Greatest Common Factor (GCF)
Look at all the coefficients and variables in each term.
- Coefficients: The coefficients are 10, -15, 20, and -30. The greatest common factor of these numbers is 5.
- Variables: The terms involve the variables [tex]\( n \)[/tex] and [tex]\( x \)[/tex]. The greatest common factor for the variables in each term is [tex]\( n \)[/tex].
So, our GCF is [tex]\( 5n \)[/tex].
### Step 2: Factor out the GCF from each term
Each term of the expression can be divided by [tex]\( 5n \)[/tex]:
[tex]\[
10n^3 \div 5n = 2n^2
\][/tex]
[tex]\[
-15n^2 \div 5n = -3n
\][/tex]
[tex]\[
20xn^2 \div 5n = 4xn
\][/tex]
[tex]\[
-30xn \div 5n = -6x
\][/tex]
### Step 3: Rewrite the expression
Using the factored-out GCF, we can rewrite the expression as:
[tex]\[
10n^3 - 15n^2 + 20xn^2 - 30xn = 5n(2n^2 - 3n + 4xn - 6x)
\][/tex]
### Step 4: Simplify the inner expression (if possible)
We need to look for a way to factor the remaining polynomial inside the parentheses. From our step-by-step approach, we notice that we can refactor further:
The given simplified form is:
[tex]\[
5n(n + 2x)(2n - 3)
\][/tex]
Thus, putting it all together, the fully factored expression is:
[tex]\[
10n^3 - 15n^2 + 20xn^2 - 30xn = 5n(n + 2x)(2n - 3)
\][/tex]
So, after factoring out the greatest common factor, the rewritten expression is [tex]\( 5n(n + 2x)(2n - 3) \)[/tex].
