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The equation provided is:

\[
4x^2 + 12x = -5
\]

### Step 1: Rearrange the equation to standard quadratic form
The quadratic formula is applied to equations in the form \( ax^2 + bx + c = 0 \). To put the given equation in this form, add 5 to both sides:

\[
4x^2 + 12x + 5 = 0
\]

Now, we have the quadratic equation in standard form, where:
- \( a = 4 \)
- \( b = 12 \)
- \( c = 5 \)

### Step 2: Apply the quadratic formula
The quadratic formula is:

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

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

\[
x = \frac{-12 \pm \sqrt{12^2 - 4(4)(5)}}{2(4)}
\]

\[
x = \frac{-12 \pm \sqrt{144 - 80}}{8}
\]

\[
x = \frac{-12 \pm \sqrt{64}}{8}
\]

\[
x = \frac{-12 \pm 8}{8}
\]

### Step 3: Solve for the two possible roots
Now, calculate the two possible roots:

1. For the positive root (\( +8 \)):

\[
x_1 = \frac{-12 + 8}{8} = \frac{-4}{8} = -\frac{1}{2}
\]

2. For the negative root (\( -8 \)):

\[
x_2 = \frac{-12 - 8}{8} = \frac{-20}{8} = -\frac{5}{2}
\]

### Final Answer:
The roots of the equation are:

\[
x = -\frac{1}{2} \quad \text{or} \quad x = -\frac{5}{2}
\]

---

Would you like further clarification on any of these steps or more details?

Here are five related questions to deepen your understanding:

1. How does the discriminant (\( b^2 - 4ac \)) help determine the number of real roots?
2. What happens if the discriminant is less than zero in the quadratic formula?
3. Can the quadratic formula be used for equations that do not have integer roots?
4. How would the solution change if we had a leading coefficient other than 1 or 4?
5. What are some other methods to solve quadratic equations besides the quadratic formula?

**Tip**: Always check the discriminant first to see whether the quadratic equation has real, repeated, or complex roots!

Solve for the roots in simplest form using the quadratic formula: 4x^2 + 12x = -5

Learn how to solve the quadratic equation 4x^2 + 12x + 5 = 0 using the quadratic formula. This problem involves key algebraic concepts and detailed steps suitable for students in Grades 8-10.

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