We are given the equation:

\[
r^2 + 2r + 14 = -32
\]

### Steps to solve:

1. **Move the constant to the other side:**

   \[
   r^2 + 2r + 14 + 32 = 0
   \]
   
   Simplifying:

   \[
   r^2 + 2r + 46 = 0
   \]

2. **Use the quadratic formula** to solve for \(r\):

   The quadratic formula is given by:

   \[
   r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]

   In this case, \(a = 1\), \(b = 2\), and \(c = 46\). Substituting these values into the formula:

   \[
   r = \frac{-2 \pm \sqrt{2^2 - 4(1)(46)}}{2(1)}
   \]

   Simplifying further:

   \[
   r = \frac{-2 \pm \sqrt{4 - 184}}{2}
   \]

   \[
   r = \frac{-2 \pm \sqrt{-180}}{2}
   \]

3. **Simplify the square root of a negative number**:

   \[
   r = \frac{-2 \pm \sqrt{-180}}{2} = \frac{-2 \pm i\sqrt{180}}{2}
   \]

   Now simplify \(\sqrt{180}\):

   \[
   \sqrt{180} = \sqrt{36 \times 5} = 6\sqrt{5}
   \]

   So the solution becomes:

   \[
   r = \frac{-2 \pm 6i\sqrt{5}}{2}
   \]

4. **Final simplified solution**:

   \[
   r = -1 \pm 3i\sqrt{5}
   \]

Thus, the solutions are:

\[
r = -1 + 3i\sqrt{5} \quad \text{or} \quad r = -1 - 3i\sqrt{5}
\]

Would you like more details on any of these steps?

### Related Questions:
1. How do you handle imaginary numbers?
2. Can we complete the square to solve this instead of using the quadratic formula?
3. What are the real-life applications of complex numbers?
4. What does the discriminant tell us about the nature of the roots?
5. How does the graph of a quadratic function relate to its solutions?

### Tip:
When the discriminant (the part inside the square root in the quadratic formula) is negative, the solutions will always involve complex numbers.

Learn how to solve the quadratic equation r^2 + 2r + 46 = 0, resulting in complex solutions using the quadratic formula. This example covers algebraic concepts, handling complex numbers, and simplifying square roots of negative numbers. Suitable for high school students in Grades 10-12.

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