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Apply substitution. Then get everything to one side.
y = 2x+1
x^2+6x+m = 2x+1
x^2+6x+m-2x-1 = 0
x^2+4x+(m-1) = 0
The last equation is of the form
ax^2+bx+c = 0
where,
a = 1
b = 4
c = m-1
Those values are plugged into the quadratic formula
The discriminant must be nonnegative so that we have real number roots (or else we won't have intersection points on the graph).
This tells us that m = 5 is the largest value of m possible so that the line and curve intersect.
If m = 5, then we have exactly one intersection point (because it makes the discriminant to be zero to point to one root only). This is when the line is tangent to the curve.
If m < 5, then we have two distinct intersection points.
The line is a secant to the curve in this case.
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