To find the value of the expression [tex]\(m^2 - 2mn + n^2\)[/tex] for [tex]\(m = -2\)[/tex] and [tex]\(n = 4\)[/tex], we will go through the steps systematically.
1. Calculate [tex]\(m^2\)[/tex]:
[tex]\[
m = -2 \implies m^2 = (-2)^2 = 4
\][/tex]
2. Calculate [tex]\(2mn\)[/tex]:
[tex]\[
m = -2, \, n = 4 \implies 2mn = 2 \cdot (-2) \cdot 4 = -16
\][/tex]
3. Calculate [tex]\(n^2\)[/tex]:
[tex]\[
n = 4 \implies n^2 = 4^2 = 16
\][/tex]
4. Combine these results into the expression [tex]\(m^2 - 2mn + n^2\)[/tex]:
[tex]\[
m^2 - 2mn + n^2 = 4 - (-16) + 16
\][/tex]
5. Simplify the expression:
[tex]\[
4 - (-16) + 16 = 4 + 16 + 16 = 36
\][/tex]
So, the value of the expression [tex]\(m^2 - 2mn + n^2\)[/tex] for [tex]\(m = -2\)[/tex] and [tex]\(n = 4\)[/tex] is [tex]\(\boxed{36}\)[/tex].
