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Let's solve the given problem step-by-step.

We need to find two tangent lines to the parabola \( y = x^2 \) that pass through the point \( (0, -4) \). Additionally, we are asked to find the points where these tangent lines intersect the parabola.

### Step 1: Equation of the Tangent Line to the Parabola
The general equation of the parabola is \( y = x^2 \). The derivative of \( y = x^2 \), which gives the slope of the tangent line at any point \( x = a \), is:

\[
\frac{dy}{dx} = 2a
\]

Thus, the slope of the tangent line at \( x = a \) is \( 2a \). The equation of the tangent line at the point \( (a, a^2) \) on the parabola is:

\[
y - a^2 = 2a(x - a)
\]
Simplifying this, we get:
\[
y = 2a(x - a) + a^2 = 2ax - a^2
\]

### Step 2: Use the Point (0, -4)
The tangent line passes through the point \( (0, -4) \). Substitute \( x = 0 \) and \( y = -4 \) into the tangent line equation \( y = 2ax - a^2 \):

\[
-4 = 2a(0) - a^2
\]
\[
-4 = -a^2
\]
\[
a^2 = 4
\]

So, \( a = 2 \) or \( a = -2 \).

### Step 3: Find the Tangent Lines and Points of Intersection
For \( a = 2 \), the equation of the tangent line is:
\[
y = 2(2)x - (2)^2 = 4x - 4
\]

For \( a = -2 \), the equation of the tangent line is:
\[
y = 2(-2)x - (-2)^2 = -4x - 4
\]

These are the two tangent lines. Now, let's find the points of intersection of these lines with the parabola \( y = x^2 \).

### Step 4: Solve for Points of Intersection
For the line \( y = 4x - 4 \) and the parabola \( y = x^2 \), equate the two equations:
\[
x^2 = 4x - 4
\]
\[
x^2 - 4x + 4 = 0
\]
\[
(x - 2)^2 = 0
\]
So, \( x = 2 \). The corresponding point is \( (2, 4) \).

For the line \( y = -4x - 4 \) and the parabola \( y = x^2 \), equate the two equations:
\[
x^2 = -4x - 4
\]
\[
x^2 + 4x + 4 = 0
\]
\[
(x + 2)^2 = 0
\]
So, \( x = -2 \). The corresponding point is \( (-2, 4) \).

### Final Answer:
The points where the tangent lines intersect the parabola are \( (2, 4) \) and \( (-2, 4) \).

Would you like further clarification on any part of the solution? 

Here are 5 related questions for further practice:
1. Find the equation of the tangent line to the parabola \( y = x^2 \) at the point \( (3, 9) \).
2. Determine the coordinates of the point(s) where the tangent line to \( y = x^2 \) at \( (1, 1) \) intersects the parabola again.
3. Find the slope of the tangent line to the curve \( y = x^3 \) at \( x = 2 \).
4. Determine the points where the tangent lines to the parabola \( y = 2x^2 \) pass through the point \( (0, -8) \).
5. For a general parabola \( y = ax^2 \), find the equation of the tangent line at \( x = c \).

**Tip**: When finding tangent lines, always remember the slope is determined by the derivative at a specific point on the curve.

Draw a diagram to show that there are two tangent lines to the parabola y = x^2 that pass through the point (0, -4). Find the coordinates of the points where these tangent lines intersect the parabola.

This problem demonstrates how to find two tangent lines to the parabola y = x^2 that pass through the point (0, -4), including the method to determine the points of intersection. Concepts include derivatives and geometry of parabolas, making it suitable for Grades 11-12.

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