Alright, let's solve the given expression step-by-step.
### Part A: Factoring out the Greatest Common Factor
First, we need to rewrite the expression [tex]\(4m^3 + 12xm^2 - 8m^2 - 24xm\)[/tex] by factoring out the greatest common factor (GCF).
1. Identify the GCF of the terms in the polynomial.
- The terms are [tex]\(4m^3\)[/tex], [tex]\(12xm^2\)[/tex], [tex]\(-8m^2\)[/tex], and [tex]\(-24xm\)[/tex].
- The numerical part shared by all coefficients is 4.
- The variable part shared by all terms is [tex]\(m\)[/tex].
Therefore, the GCF of the expression is [tex]\(4m\)[/tex].
2. Factor out the GCF from each term:
- [tex]\(4m^3\)[/tex] becomes [tex]\(4m \cdot m^2\)[/tex]
- [tex]\(12xm^2\)[/tex] becomes [tex]\(4m \cdot 3xm\)[/tex]
- [tex]\(-8m^2\)[/tex] becomes [tex]\(4m \cdot -2m\)[/tex]
- [tex]\(-24xm\)[/tex] becomes [tex]\(4m \cdot -6x\)[/tex]
So, the rewritten expression by factoring out the GCF is:
[tex]\[ 4m(m^2 + 3xm - 2m - 6x) \][/tex]
### Part B: Factoring the Expression Completely
Next, we need to factor the expression [tex]\( m^2 + 3xm - 2m - 6x \)[/tex] completely.
1. Group the terms to simplify:
- [tex]\((m^2 + 3xm) - (2m + 6x)\)[/tex]
2. Factor out the common factors from each group:
- [tex]\(m(m + 3x)\)[/tex] from the first group [tex]\((m^2 + 3xm)\)[/tex]
- [tex]\(-2(m + 3x)\)[/tex] from the second group [tex]\((-2m - 6x)\)[/tex]
So, the expression becomes:
[tex]\[ m(m + 3x) - 2(m + 3x) \][/tex]
3. Notice that [tex]\((m + 3x)\)[/tex] is a common factor:
- Factor out [tex]\((m + 3x)\)[/tex]:
- [tex]\((m + 3x)(m - 2)\)[/tex]
Combining both steps, we have:
[tex]\[ 4m(m + 3x)(m - 2) \][/tex]
### Complete Solution
- Part A: The expression with the greatest common factor factored out is:
[tex]\[ 4m(m^2 + 3xm - 2m - 6x) \][/tex]
- Part B: The completely factored expression is:
[tex]\[ 4m(m + 3x)(m - 2) \][/tex]
The complete solutions to both parts are intact and correct as presented.
