We are given the equation:

\[
x(x + 4) = -5
\]

Let's solve it step by step:

### Step 1: Expand the left side.
Distribute \( x \) across the terms inside the parentheses:

\[
x^2 + 4x = -5
\]

### Step 2: Rearrange the equation.
Move all terms to one side to set the equation equal to zero:

\[
x^2 + 4x + 5 = 0
\]

### Step 3: Solve the quadratic equation.
We will now solve this quadratic equation using the **quadratic formula**. The general form of a quadratic equation is:

\[
ax^2 + bx + c = 0
\]

For our equation \( x^2 + 4x + 5 = 0 \), we have:
- \( a = 1 \)
- \( b = 4 \)
- \( c = 5 \)

The quadratic formula is:

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

Substitute the values of \( a \), \( b \), and \( c \):

\[
x = \frac{-4 \pm \sqrt{4^2 - 4(1)(5)}}{2(1)}
\]

Simplify inside the square root:

\[
x = \frac{-4 \pm \sqrt{16 - 20}}{2}
\]

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

Since the discriminant is negative, we have complex solutions. The square root of \(-4\) is \( 2i \), where \( i \) is the imaginary unit. So we get:

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

### Step 4: Simplify the solutions.
Simplify the expression:

\[
x = -2 \pm i
\]

Thus, the two complex solutions are:

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

### Final Answer:
The solutions are \( x = -2 + i \) and \( x = -2 - i \).

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. What is the quadratic formula and when is it used?
2. How do you handle negative discriminants in quadratic equations?
3. What is the significance of complex numbers in mathematics?
4. How can you graph equations that have complex solutions?
5. What are some real-world applications of quadratic equations?

**Tip:** When the discriminant of a quadratic equation is negative, the solutions will always involve complex numbers (with \( i \)).

Discover how to solve the quadratic equation x(x + 4) = -5 using the quadratic formula, leading to complex solutions. This detailed, step-by-step guide covers key algebraic concepts and is suitable for students in Grades 9-12.

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