https://file.aiquickdraw.com/mathbot/file/1726498479348szf59p38.jpg

The equation in the image is:

\[
(3x^2 + 3x - 6)(x^2 + 4) = 0
\]

To solve this, we need to apply the zero product property, which states that if the product of two expressions is zero, then at least one of the expressions must be zero.

### Step 1: Solve \(3x^2 + 3x - 6 = 0\)

We can simplify this quadratic by dividing through by 3:

\[
x^2 + x - 2 = 0
\]

Now, solve this quadratic equation using factoring. We need to find two numbers that multiply to \(-2\) and add to \(1\). The factors are \(2\) and \(-1\), so we factor the quadratic as:

\[
(x - 1)(x + 2) = 0
\]

Thus, the solutions for this part are:
\[
x = 1 \quad \text{or} \quad x = -2
\]

### Step 2: Solve \(x^2 + 4 = 0\)

This is a simple quadratic equation. Rearrange it:

\[
x^2 = -4
\]

Taking the square root of both sides:

\[
x = \pm \sqrt{-4} = \pm 2i
\]

Here, \(i\) is the imaginary unit, so the solutions are:

\[
x = 2i \quad \text{or} \quad x = -2i
\]

### Final Solution:

The complete set of solutions is:

\[
x = 1, \, x = -2, \, x = 2i, \, x = -2i
\]

Would you like more details, or have any questions?

Here are 5 related questions to explore further:
1. How do you solve a quadratic equation using the quadratic formula?
2. What are the properties of imaginary numbers and how are they used?
3. Can the quadratic formula be used to solve \(x^2 + 4 = 0\)?
4. What happens if we graph \(3x^2 + 3x - 6\) on a coordinate plane?
5. How does factoring differ from using the quadratic formula?

**Tip:** Always check if an equation can be factored before using more advanced methods like the quadratic formula; it can save time!

Learn how to solve the quadratic equation (3x^2 + 3x - 6)(x^2 + 4) = 0 with a step-by-step solution. This solution involves applying the zero product property, factoring, and solving for both real and imaginary solutions. Suitable for high school students in Grades 10-12.

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