Please explain how do you solve these kind of problems

Find the set of values of m for which the line y = mx + 4 intersects the curve y = 3x^2 − 4x + 7 at two distinct points.

thanks in advance ^_^

At the points of intersection, the y values must be equal. So

mx + 4 = 3x² - 4x + 7
0 = 3x² - (m+4)x + 3

For the general quadratic equation ax² + bx + c = 0 there is a quantity called the discriminant D:

D = b² - 4ac

If D > 0, there are two distinct real solutions
If D = 0, there is one real solution (a double root)
If D < 0, there are no real solutions

For your equation,

D = (-(m+4))² - 4*3*3
  = m² + 8m + 16 - 36
  = m² + 8m - 20

So you want m such that m² + 8m - 20 > 0

First, find the solutions of m² + 8m - 20 = 0:

m² + 8m - 20 = 0
(m + 10)(m - 2) = 0
m = -10 or m = 2

These two points divide the number line into three regions within each of which the sign of m² + 8m - 20 is constant: m < -10, -10 < m < 2, and m > 2. So choose a value in each of the three regions and evaluate m² + 8m - 20 at that value. Its sign tells you the sign of m² + 8m - 20 throughout that region.

m < -10: Try m = -11. (-11)² + 8(-11) - 20 > 0
-10 < m < 2: Try m = 0: (0)² + 8(0) - 20 < 0
m > 2: Try m = 3: (3)² + 8(3) - 20 > 0

So the line y = mx + 4 intersects the curve (a parabola, by the way) 3x² - 4x + 7 at two distinct points if m < -10 or m > 2

The line and curve will intersect when they are equal:

3x^2 - 4x + 7 = mx + 4. Move everything to the left side:

3x^2 + (-4 - m)*x + 3 = 0.

We need to know when this equation has two solutions. Because it is quadratic, we know it will happen when the discriminant is greater than 0:

a = 3;
b = -4 - m;
c = 3.

b^2 - 4ac = (-4 - m)^2 - 4*3*3
= m^2 + 8m + 16 - 36 = m^2 + 8m - 20 = (m+10)*(m-2).

Now, you need to solve (m+10)*(m-2) > 0. I'll leave that to you.