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We are tasked with solving the following equation:

\[
(-7 - 4x)^2 = 3
\]

To solve this, follow these steps:

### Step 1: Take the square root of both sides
Since we have a square on the left-hand side, we can remove the square by taking the square root of both sides:

\[
\sqrt{(-7 - 4x)^2} = \pm \sqrt{3}
\]

This simplifies to:

\[
-7 - 4x = \pm \sqrt{3}
\]

### Step 2: Solve for \( x \)

#### Case 1: \( -7 - 4x = \sqrt{3} \)
Solve for \( x \):
\[
-4x = \sqrt{3} + 7
\]
\[
x = \frac{7 + \sqrt{3}}{-4}
\]

#### Case 2: \( -7 - 4x = -\sqrt{3} \)
Solve for \( x \):
\[
-4x = -\sqrt{3} + 7
\]
\[
x = \frac{7 - \sqrt{3}}{-4}
\]

### Step 3: Express the solution as a set

The solutions for \( x \) are:

\[
x = \left\{\frac{7 + \sqrt{3}}{-4}, \frac{7 - \sqrt{3}}{-4}\right\}
\]

Thus, the solution set is:
\[
\boxed{\left\{\frac{7 + \sqrt{3}}{-4}, \frac{7 - \sqrt{3}}{-4}\right\}}
\]

Would you like a more detailed explanation or help with any part of the solution? Here are some related questions you could explore:

1. What are the steps for solving quadratic equations?
2. How do we handle square roots when solving equations?
3. How do we express sets of solutions for equations like this?
4. Can this type of equation have no real solutions?
5. How can we check if our solutions are correct?

**Tip:** Always check the solutions by substituting them back into the original equation to verify.

(-7 - 4x)^2 = 3. Solve for x and express the solution as a set.

Learn how to solve the equation (-7 - 4x)^2 = 3 step by step by taking the square root of both sides. This solution involves key algebraic concepts and solving techniques using square roots. Suitable for students in Grades 9-11.

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