Find what values of m is the line 
a tangent to the parabola 
?
Here is the parabola and the tangent line we want:
We solve this system to find their common point:
by substitution:
4 - x² = mx + 5
-x² - mx - 1 = 0
x² + mx + 1 = 0
This must have exactly ONE real solution, for if it has
TWO real solutions the line will cut the parabola in TWO
points, and not be tangent, like this:
Or if it has NO real solutions but only imaginary solutions
it will not touch the parabola at all, like this:
So to get it tangent to the parabola, we must be sure that
x² + mx + 1 = 0
has only ONE solution for x. That means its discriminant b²-4ac must
equal to 0, so
a = 1, b = m, c = 1
discriminant = b² - 4ac = m² - 4(1)(1) = m² - 4
This discriminant must = 0, so
m² - 4 = 0
(m - 2)(m + 2) = 0
m - 2 = 0; m + 2 = 0
m = 2; m = -2
So there are two possible values for m, which means there are TWO lines
of the form y = mx + 5 which will be tangent to the parabola. They are
y = 2x + 5 and y = -2x + 5
Answer: m = 2, m = -2
Edwin
