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To solve the problem and find the value of \( c \) such that the function \( f(x) \) is continuous at \( x = 4 \), we must ensure that the left-hand limit of \( f(x) \) as \( x \to 4^- \) equals the right-hand limit as \( x \to 4^+ \), and also that these limits equal the value of \( f(4) \).

### Step 1: Analyze the given function
The function \( f(x) \) is defined piecewise as:
\[
f(x) = 
\begin{cases} 
c^2 x^2 & \text{if } x < 4 \\
16cx + 80 & \text{if } x \geq 4
\end{cases}
\]

To ensure continuity at \( x = 4 \), the following condition must hold:
\[
\lim_{x \to 4^-} f(x) = \lim_{x \to 4^+} f(x) = f(4)
\]

### Step 2: Left-hand limit (\( x \to 4^- \))
For \( x < 4 \), we have \( f(x) = c^2 x^2 \). So, the left-hand limit as \( x \to 4^- \) is:
\[
\lim_{x \to 4^-} f(x) = \lim_{x \to 4^-} c^2 x^2 = c^2 \cdot 4^2 = 16c^2
\]

### Step 3: Right-hand limit (\( x \to 4^+ \))
For \( x \geq 4 \), we have \( f(x) = 16cx + 80 \). So, the right-hand limit as \( x \to 4^+ \) is:
\[
\lim_{x \to 4^+} f(x) = 16c \cdot 4 + 80 = 64c + 80
\]

### Step 4: Continuity condition
For continuity at \( x = 4 \), we set the left-hand limit equal to the right-hand limit:
\[
16c^2 = 64c + 80
\]

### Step 5: Solve the equation
We now solve the quadratic equation:
\[
16c^2 - 64c - 80 = 0
\]
Divide through by 16 to simplify:
\[
c^2 - 4c - 5 = 0
\]
Factor the quadratic equation:
\[
(c - 5)(c + 1) = 0
\]
So, the solutions are:
\[
c = 5 \quad \text{or} \quad c = -1
\]

Thus, the values of \( c \) that make \( f(x) \) continuous at \( x = 4 \) are \( c = 5 \) and \( c = -1 \).

### Final Answer:
\[
c = 5, -1
\]

Would you like more details on any specific part of this solution?

Here are some related questions to explore:
1. What happens to the function if \( c \) takes any other value?
2. How does differentiability at \( x = 4 \) compare with continuity for this function?
3. Can you solve for continuity at other points (besides \( x = 4 \))?
4. How does the quadratic factorization method work?
5. What are the general conditions for piecewise functions to be continuous?

**Tip**: When solving continuity problems, always set the left-hand and right-hand limits equal and make sure they match the function's value at the given point.

Consider the function f(x) = { c^2 * x^2, if x < 4; 16cx + 80, if x ≥ 4 } where c is a constant. Find all value(s) of the constant c such that f(x) is continuous at x = 4. List the values of c that you find separated by commas.

Explore how to determine the continuity of a piecewise function at x = 4 by finding the appropriate value of the constant c. This problem involves analyzing limits and solving a quadratic equation, suitable for high school students in Grades 10-12.

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