Let's solve the equation step by step:

The equation is:

$x^2 + 6x = -\frac{11}{4}$

First, let's move everything to one side of the equation to make it easier to work with:

$x^2 + 6x + \frac{11}{4} = 0$

Now, let's eliminate the fraction by multiplying the whole equation by 4:

$4x^2 + 24x + 11 = 0$

This is now a quadratic equation in standard form. We can solve it using the quadratic formula, which is:

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

For the equation $4x^2 + 24x + 11 = 0$, the coefficients are:

• $a = 4$
• $b = 24$
• $c = 11$

Now, we can plug these values into the quadratic formula:

$x = \frac{-24 \pm \sqrt{24^2 - 4 \cdot 4 \cdot 11}}{2 \cdot 4}$

Simplifying further:

$x = \frac{-24 \pm \sqrt{576 - 176}}{8}$ $x = \frac{-24 \pm \sqrt{400}}{8}$ $x = \frac{-24 \pm 20}{8}$

Now, we can solve for both possible values of $x$:

1. $x_1 = \frac{-24 + 20}{8} = \frac{-4}{8} = -\frac{1}{2}$
2. $x_2 = \frac{-24 - 20}{8} = \frac{-44}{8} = -\frac{11}{2}$

So the solutions are:

$x = -\frac{1}{2} \quad \text{and} \quad x = -\frac{11}{2}$

Looking at the answer choices, the correct option is D.

Would you like further details or clarifications on this solution?

Here are 5 follow-up questions to expand on this:

1. How does multiplying through by 4 simplify the equation in this case?
2. Why do we use the quadratic formula to solve this equation?
3. Can you factor the quadratic equation instead of using the formula?
4. How can you check if your solutions are correct by substitution?
5. What is the significance of the discriminant $\Delta = b^2 - 4ac$ in this problem?

Tip: Always simplify fractions or equations early on to make solving quadratic equations easier.