To factor the given expression, [tex]\(16m^2n + 48m^4 - 8m^3n\)[/tex], we start by identifying the greatest common factor (GCF) of the terms:
1. Identify the coefficients and variables:
- For [tex]\(16m^2n\)[/tex], the coefficient is 16, and the variables are [tex]\(m^2\)[/tex] and [tex]\(n\)[/tex].
- For [tex]\(48m^4\)[/tex], the coefficient is 48, and the variable is [tex]\(m^4\)[/tex].
- For [tex]\(8m^3n\)[/tex], the coefficient is 8, and the variables are [tex]\(m^3\)[/tex] and [tex]\(n\)[/tex].
2. Find the GCF of the coefficients:
- The coefficients are 16, 48, and 8.
- The factors of 16 are [tex]\(1, 2, 4, 8, 16\)[/tex].
- The factors of 48 are [tex]\(1, 2, 3, 4, 6, 8, 12, 16, 24, 48\)[/tex].
- The factors of 8 are [tex]\(1, 2, 4, 8\)[/tex].
- The GCF of 16, 48, and 8 is 8.
3. Find the common variables:
- The variable with the lowest power for [tex]\(m\)[/tex] is [tex]\(m^2\)[/tex] (since [tex]\(m^2\)[/tex], [tex]\(m^4\)[/tex], and [tex]\(m^3\)[/tex] are present).
- The variable [tex]\(n\)[/tex] is present in two of the three terms.
4. Combine the common coefficients and variables:
- The GCF is [tex]\(8m^2\)[/tex] (we take [tex]\(m^2\)[/tex] because it is the lowest power of [tex]\(m\)[/tex] present in all terms).
5. Factor out the GCF from each term:
[tex]\[
16m^2n + 48m^4 - 8m^3n = 8m^2 \cdot \left(\frac{16m^2n}{8m^2} + \frac{48m^4}{8m^2} - \frac{8m^3n}{8m^2}\right)
\][/tex]
6. Simplify the expression inside the parentheses:
[tex]\[
= 8m^2 \left(2n + 6m^2 - mn\right)
\][/tex]
Therefore, the factored form of the expression [tex]\(16m^2n + 48m^4 - 8m^3n\)[/tex] is:
[tex]\[
8m^2(6m^2 - mn + 2n)
\][/tex]
