Sometimes true, always true, or never true?

Explanation:

I'll use x in place of n

Let y = x^2 - 4x + 5

If we complete the square, then,

y = x^2 - 4x + 5

y = (x^2 - 4x) + 5

y = (x^2 - 4x + 4 - 4) + 5

y = (x^2 - 4x + 4) - 4 + 5

y = (x-2)^2 + 1

The quantity (x-2)^2 is never negative as squaring any real number value is never a negative result. Adding on 1 makes the result positive. So y > 0 regardless of whatever x is. Replace x with n, and this shows how n^2 - 4n + 5 is always positive for any integer n.

You could also use the quadratic formula to find that x^2 - 4x + 5 = 0 has no real solutions, so there are no x intercepts. Either the graph is entirely above the x axis or it is entirely below the x axis.

Plug in any x value you want, say x = 0, and the result is positive. Meaning that whatever x value you plug in will be positive (as the graph can't cross the x axis to go into negative territory)

the quotient is 4x^2 - 27x + 167 with a remainder of -996

step-by-step explanation:

in synthetic division, use -6 as the divisor and use the coefficients {4, -3, 5, 6}:

-6     )   4     -3     5     6

                -24   162 -1002

          4   -27   167   -996

thus, the quotient is 4x^2 - 27x + 167 with a remainder of -996.

note:   your "4x°-3x+5x+6" is ambiguous / incomplete.   i had to assume that you meant '4x^3 - 3x^2 + 5x + 6."

think about the doctor who treated the major and the speaker in the short story "in another country." put yourself in the shoes of the doctor, and based on what you read in the story, write what you think his thoughts about the major and the speaker would be. use the following vocabulary in your response and change the form of these words as needed: detached resign withered