To find the value of k, we need to set the two given expressions equal to each other and solve for k.
First, let's expand the first expression: 2(2²-42-21)-(z-7)(z+77) = 4 - 8 - 42 - 21 - (z-7)(z+77) = -47 - (z-7)(z+77)
Now, let's set this equal to the second expression: -47 - (z-7)(z+77) = (z-7)(z+k)
Expanding the second expression, we get: -47 - (z-7)(z+k) = z^2 - 7z + zk - 7k
Matching the coefficients of like terms, we get:
-47 = z^2 - 7z + zk - 7k
-47 = z^2 + zk - 7z - 7k
0 = z^2 + zk - 7z - 7k + 47
0 = (z-7)(z+k) + 47
Since the expression in parentheses is equal to zero, we have:
0 = 47
This equation has no solutions, so the given expressions are not equivalent. This means that we cannot find a value of k that makes the two expressions equal.
Therefore, the answer is none of the given options.