Graph y = −x^2 + 4x + 5 by plotting the solutions of the equation when x = −2, −1, 0, 1, 2, 3, and 4?

y = -x² + 4x + 5

When x = -2:
y = -(-2)² + 4(-2) + 5
y = -4 - 8 + 5
y = -7

When x = -1:
y = -(-1)² + 4(-1) + 5
y = -1 - 4 + 5
y = 0

When x = 0:
y = -(0)² + 4(0) + 5
y = 0 + 0 + 5
y = 5

When x = 1:
y = -(1)² + 4(1) + 5
y = -1 + 4 + 5
y = 8

When x = 2:
y = -(2)² + 4(2) + 5
y = -4 + 8 + 5
y = 9

When x = 3:
y = -(3)² + 4(3) + 5
y = -9 + 12 + 5
y = 8

When x = 4:
y = -(4)² + 4(4) + 5
y = -16 + 16 + 5
y = 5

A table of values can be found at the source link.

I'll do the first and last. Then compare that to what you did and if you did something wrong, you'll see why.

y = -x² + 4x + 5
y = -(-2)² + 4(-2) + 5 and y = -4² + 4(4) + 5
y = -4 - 8 + 5 and y = -16 + 16 + 5
y = -7 and y = 5

Remember that the minus is not squared .

-x² will always result in a negative number (other than x = 0)

different from (-x)² where the result will always be positive (other than x = 0)