Solve the inequality. Write the solution set in interval notation if possible. Simplify your answer, if necessary.

<+4a28a5+a2
The solution set is 
 
.


It looks like there might be a typo in your inequality. Based on what you wrote, "<+4a28a5+a2" doesn't form a clear mathematical inequality. I assume you meant something along the lines of:

\[
4a^2 + 8a + 5 + a^2 < 0
\]

Let's solve this inequality. First, combine like terms:

\[
4a^2 + a^2 + 8a + 5 < 0
\]

\[
5a^2 + 8a + 5 < 0
\]

This is a quadratic inequality. To solve it, we can proceed by finding the roots of the quadratic equation:

\[
5a^2 + 8a + 5 = 0
\]

Using the quadratic formula \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 5\), \(b = 8\), and \(c = 5\):

\[
a = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 5 \cdot 5}}{2 \cdot 5}
\]

\[
a = \frac{-8 \pm \sqrt{64 - 100}}{10}
\]

\[
a = \frac{-8 \pm \sqrt{-36}}{10}
\]

Since the discriminant (\(64 - 100 = -36\)) is negative, the quadratic equation has no real roots. This means that the quadratic expression \(5a^2 + 8a + 5\) does not cross the x-axis and is always positive (or negative) depending on the sign of the leading coefficient.

Given that the leading coefficient \(5\) is positive, the quadratic \(5a^2 + 8a + 5\) is always positive for all real \(a\). Therefore, it is impossible for \(5a^2 + 8a + 5 < 0\) to be true for any real number \(a\).

Thus, the solution set is:

\[
\text{The solution set is } \emptyset \text{ (the empty set)}.
\]

Would you like more details, or do you have any questions? Here are some related questions to further explore:

1. How do we determine the nature of the roots of a quadratic equation?
2. What does it mean for a quadratic inequality to have no real solution?
3. Can you explain how the discriminant affects the shape of the parabola for a quadratic function?
4. What are the steps to solve quadratic inequalities that do have real solutions?
5. How can we graphically represent the solution set of a quadratic inequality?

**Tip:** When solving quadratic inequalities, always start by finding the roots of the corresponding quadratic equation. The nature of the roots (real or complex) will give insights into the inequality's solution.

Learn how to solve the quadratic inequality 5a^2 + 8a + 5 < 0 with a detailed step-by-step solution. This problem involves quadratic inequalities and the quadratic formula, suitable for high school students.

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