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The problem asks to find the values of \( a \) where the line tangent to the graph of \( f(x) = 8x^3 + 60x^2 + 146x + 72 \) at \( x = a \) is parallel to the line \( y = 2x + 0.4 \).

### Step-by-Step Solution

1. **Derivative of \( f(x) \):**
   To find where the tangent line is parallel to the given line, we need the derivative of \( f(x) \), because the slope of the tangent line at any point \( x \) is given by \( f'(x) \). So, let's find \( f'(x) \).

   \[
   f(x) = 8x^3 + 60x^2 + 146x + 72
   \]
   Differentiate \( f(x) \):
   \[
   f'(x) = 24x^2 + 120x + 146
   \]

2. **Slope condition:**
   The tangent line will be parallel to the line \( y = 2x + 0.4 \), which has a slope of 2. Therefore, we want to find the values of \( x \) (or \( a \)) where \( f'(x) = 2 \).

   So, solve the equation:
   \[
   24x^2 + 120x + 146 = 2
   \]

3. **Simplify the equation:**
   Subtract 2 from both sides:
   \[
   24x^2 + 120x + 144 = 0
   \]
   Simplify by dividing the entire equation by 24:
   \[
   x^2 + 5x + 6 = 0
   \]

4. **Solve the quadratic equation:**
   Factor the quadratic equation:
   \[
   (x + 2)(x + 3) = 0
   \]
   Therefore, the solutions are:
   \[
   x = -2 \quad \text{or} \quad x = -3
   \]

Thus, the values of \( a \) are \( a = -2 \) and \( a = -3 \).

Would you like more details or have any questions?

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Here are five follow-up questions:
1. How is the derivative related to the slope of the tangent line?
2. Why is the slope of the line \( y = 2x + 0.4 \) equal to 2?
3. What is the general process for finding where a tangent line is parallel to another line?
4. How would the result change if the equation of the line was different, for example, \( y = 3x + 1 \)?
5. What are the key steps involved in solving a quadratic equation?

**Tip:** When dealing with polynomials, remember to use factoring, completing the square, or the quadratic formula to solve quadratic equations efficiently.

For what value(s) a is the line tangent to the graph of f(x) = 8x^3 + 60x^2 + 146x + 72 at x = a parallel to the line y = 2x + 0.4?

Learn how to solve for the values of 'a' where the tangent line to f(x) = 8x^3 + 60x^2 + 146x + 72 is parallel to the line y = 2x + 0.4. This calculus problem involves finding the derivative, solving quadratic equations, and understanding the concept of parallel lines.

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