To find integers that make the equation \( (n - k + 2)^2 \) equal to the right side, we first simplify the left-hand side:
\[ (n - k + 2)^2 = (n - k)^2 + 2(n - k)(2) + 2^2 \] Expanding \( (n - k)^2 \):
\[ (n - k)^2 = n^2 - 2nk + k^2 \]
Now, expanding \( 2(n - k)(2) \):
\[ 2(n - k)(2) = 4(n - k) = 4n - 4k \]
Finally, we add \( 2^2 = 4 \) to the expression:
\[ (n - k + 2)^2 = n^2 - 2nk + k^2 + 4n - 4k + 4 \]
Now, we can rewrite it as:
\[ (n - k + 2)^2 = n^2 + k^2 - 2nk + 4n - 4k + 4 \]
Now, we can compare coefficients with the form:
\[ (n - k + 2)^2 = __ n^2 + __ k^2 + __ n + __ k + __ nk + __ \]
By comparing, we find:
- Coefficient of \( n^2 \): 1
- Coefficient of \( k^2 \): 1
- Coefficient of \( n \): 4
- Coefficient of \( k \): -4
- Coefficient of \( nk \): -2
- Constant term: 4
Putting it all together:
\[ (n - k + 2)^2 = 1 n^2 + 1 k^2 + 4 n - 4 k - 2 nk + 4 \]
Thus, the completed integers are:
1, 1, 4, -4, -2, 4.
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