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We are tasked with solving the inequality:

\[
x^2 + 10x + 27 \geq 0
\]

### Step 1: Analyze the quadratic expression
We need to check whether the quadratic expression has any real roots by calculating its discriminant (\(\Delta\)) using the formula:

\[
\Delta = b^2 - 4ac
\]

For the equation \(x^2 + 10x + 27\), the coefficients are:
- \(a = 1\),
- \(b = 10\),
- \(c = 27\).

Thus:

\[
\Delta = 10^2 - 4(1)(27) = 100 - 108 = -8
\]

### Step 2: Conclusion from the discriminant
Since the discriminant is negative (\(\Delta = -8\)), there are no real roots. The quadratic equation does not intersect the x-axis, and the parabola is always either entirely above or below the x-axis.

### Step 3: Direction of the parabola
The coefficient of \(x^2\) (which is \(a = 1\)) is positive, meaning that the parabola opens upwards. Since there are no real roots, and the parabola opens upwards, it implies that the quadratic expression \(x^2 + 10x + 27\) is always positive for all \(x\).

### Step 4: Solving the inequality
Because \(x^2 + 10x + 27 > 0\) for all \(x \in \mathbb{R}\), the solution to the inequality:

\[
x^2 + 10x + 27 \geq 0
\]

is all real numbers, i.e., \(x \in \mathbb{R}\).

### Step 5: Answer choice
From the given options:
- **a) \(x \in \mathbb{R}\)** is the correct answer.

Would you like more details or have any questions about this solution?

Here are 5 related questions you might want to explore:
1. How do you solve a quadratic inequality with real roots?
2. What happens if the parabola opens downwards in similar problems?
3. Can you explain how the discriminant helps determine the number of real roots?
4. How would the solution change if the inequality was \(x^2 + 10x + 27 \leq 0\)?
5. What are the steps to factor a quadratic expression?

**Tip:** When solving quadratic inequalities, always check the discriminant first to determine the nature of the roots.

Solve the inequality: x^2 + 10x + 27 ≥ 0.

Learn how to solve the quadratic inequality x^2 + 10x + 27 ≥ 0 with a step-by-step explanation. Understand the use of the discriminant to determine the nature of the roots and why the solution covers all real numbers. Suitable for students in Grades 9-10.

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