Given the quadratic equation $4x^2 + 12x + 5 = 0$, we need to find the two possible values of $x$ using the quadratic formula:

The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = 4$, $b = 12$, and $c = 5$. Let's plug these values into the formula:

Step 1: Calculate the Discriminant

$b^2 - 4ac = 12^2 - 4 \times 4 \times 5 = 144 - 80 = 64$

Step 2: Find the Square Root of the Discriminant

$\sqrt{64} = 8$

Step 3: Apply the Quadratic Formula

$x = \frac{-12 \pm 8}{2 \times 4} = \frac{-12 \pm 8}{8}$

Step 4: Calculate the Two Possible Values of $x$

1. For the plus sign: $x_1 = \frac{-12 + 8}{8} = \frac{-4}{8} = -0.5$

2. For the minus sign: $x_2 = \frac{-12 - 8}{8} = \frac{-20}{8} = -2.5$

Final Answer

The two possible values of $x$ are: $x_1 = -0.5 \quad \text{and} \quad x_2 = -2.5$

Would you like more details, or do you have any questions?

Here are 5 related questions you might find helpful:

1. How do you know when a quadratic equation has real solutions?
2. What does the discriminant tell you about the nature of the roots of a quadratic equation?
3. Can the quadratic formula be used for all quadratic equations?
4. How do you factorize a quadratic equation when it's factorable?
5. What are other methods for solving quadratic equations besides the quadratic formula?

Tip: Always check the discriminant ($b^2 - 4ac$) before using the quadratic formula. It helps you identify the nature of the roots: real or complex.