Sure, let's solve the expression step-by-step:
We are given the expression:
[tex]\[
(4m - 5n)(16m^2 + 20mn + 25m^2)
\][/tex]
First, let's simplify the expression by combining like terms inside the parentheses:
[tex]\[
16m^2 + 25m^2 + 20mn = 41m^2 + 20mn
\][/tex]
So, the expression becomes:
[tex]\[
(4m - 5n)(41m^2 + 20mn)
\][/tex]
Now, let's use the distributive property (FOIL method) to expand the expression.
1. Distribute [tex]\(4m\)[/tex] across the terms inside the parentheses:
[tex]\[
4m \cdot 41m^2 + 4m \cdot 20mn = 164m^3 + 80m^2n
\][/tex]
2. Distribute [tex]\(-5n\)[/tex] across the terms inside the parentheses:
[tex]\[
-5n \cdot 41m^2 + -5n \cdot 20mn = -205m^2n - 100mn^2
\][/tex]
3. Combine all the terms:
[tex]\[
164m^3 + 80m^2n - 205m^2n - 100mn^2
\][/tex]
4. Combine like terms ([tex]\(80m^2n - 205m^2n\)[/tex]):
[tex]\[
164m^3 + (80m^2n - 205m^2n) - 100mn^2 = 164m^3 - 125m^2n - 100mn^2
\][/tex]
So, the expanded form of the expression [tex]\((4m - 5n)(41m^2 + 20mn)\)[/tex] is:
[tex]\[
164 m^3 - 125 m^2 n - 100 mn^2
\][/tex]
This is the desired result.
