To find the values of $m$ such that the line $y = mx - 4$ intersects the curve $y = -x^2 + 6x - 8$ at two distinct points, we need to solve the system of equations and analyze the discriminant of the resulting quadratic equation.

Step 1: Set the two equations equal

We equate the two equations because they represent the points of intersection:

$mx - 4 = -x^2 + 6x - 8$

Step 2: Rearrange the equation into a quadratic form

Move all terms to one side to form a quadratic equation:

$x^2 + (m - 6)x + (-4 + 8) = 0$

Simplifying:

$x^2 + (m - 6)x + 4 = 0$

This is the quadratic equation:

$x^2 + (m - 6)x + 4 = 0$

Step 3: Apply the discriminant condition

For the line to intersect the curve at two distinct points, the discriminant of the quadratic equation must be positive. The discriminant ($\Delta$) for a quadratic equation $ax^2 + bx + c = 0$ is given by:

$\Delta = b^2 - 4ac$

Here, $a = 1$, $b = m - 6$, and $c = 4$. Substituting these values into the discriminant formula:

$\Delta = (m - 6)^2 - 4(1)(4)$ $\Delta = (m - 6)^2 - 16$

For two distinct points, the discriminant must be positive:

$(m - 6)^2 - 16 > 0$

Step 4: Solve the inequality

Solve $(m - 6)^2 - 16 > 0$:

$(m - 6)^2 > 16$

Take the square root of both sides:

$|m - 6| > 4$

This gives two conditions:

$m - 6 > 4 \quad \text{or} \quad m - 6 < -4$

Solving each:

1. $m > 10$
2. $m < 2$

Step 5: Conclusion

The values of $m$ such that the line intersects the curve at two distinct points are:

$m < 2 \quad \text{or} \quad m > 10$

Would you like further details or clarifications on any part of this?

Here are some related questions:

1. What happens if $m$ is between 2 and 10?
2. How do you determine if a line is tangent to a curve?
3. How would you solve if the discriminant equals zero?
4. Can the quadratic formula be used to find exact points of intersection?
5. What is the geometric interpretation of $m$?

Tip: Always check the discriminant for any quadratic equation when analyzing intersections, as it tells you the number of solutions (intersections).