Let's solve the quadratic equation:

\[
3x^2 + 2x = -5
\]

### Step 1: Move all terms to one side of the equation

\[
3x^2 + 2x + 5 = 0
\]

### Step 2: Use the quadratic formula

The quadratic formula is given by:

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

For the equation \(3x^2 + 2x + 5 = 0\), the coefficients are:
- \(a = 3\)
- \(b = 2\)
- \(c = 5\)

### Step 3: Calculate the discriminant

\[
\Delta = b^2 - 4ac = 2^2 - 4(3)(5) = 4 - 60 = -56
\]

Since the discriminant is negative, the solutions will be complex.

### Step 4: Solve for \(x\)

\[
x = \frac{-2 \pm \sqrt{-56}}{2(3)} = \frac{-2 \pm \sqrt{56}i}{6}
\]

Simplify further:

\[
x = \frac{-2 \pm 2\sqrt{14}i}{6} = \frac{-1 \pm \sqrt{14}i}{3}
\]

### Final answer:
The solutions are:

\[
x = \frac{-1 + \sqrt{14}i}{3}, \quad x = \frac{-1 - \sqrt{14}i}{3}
\]

These are the two complex solutions.

Would you like more details on any of the steps, or do you have any other questions?

Here are some related questions:
1. How do you find real solutions for a quadratic equation?
2. What happens when the discriminant is positive?
3. Can you explain how complex numbers are used in equations?
4. How can you graph the solutions to a quadratic equation?
5. How do you solve quadratic equations by completing the square?

**Tip**: A negative discriminant indicates that the quadratic has no real solutions, only complex ones.

Learn how to solve the quadratic equation 3x^2 + 2x = -5 using the quadratic formula. This problem involves calculating the discriminant, identifying complex solutions, and simplifying the results. Suitable for students in Grades 9-12 studying algebra and complex numbers.

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