Thank you for the correction. The inequality you provided is:

\[
4a^2 + 8a < 5(a + 2)
\]

Let's solve this step-by-step:

1. **Expand and simplify the inequality**:

   \[
   4a^2 + 8a < 5(a + 2)
   \]

   Distribute \(5\) on the right-hand side:

   \[
   4a^2 + 8a < 5a + 10
   \]

2. **Move all terms to one side to set the inequality to zero**:

   \[
   4a^2 + 8a - 5a - 10 < 0
   \]

   Combine like terms:

   \[
   4a^2 + 3a - 10 < 0
   \]

3. **Solve the quadratic inequality**:

   First, solve the corresponding quadratic equation:

   \[
   4a^2 + 3a - 10 = 0
   \]

   Use the quadratic formula:

   \[
   a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]

   Here, \(a = 4\), \(b = 3\), and \(c = -10\):

   \[
   a = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 4 \cdot (-10)}}{2 \cdot 4}
   \]

   \[
   a = \frac{-3 \pm \sqrt{9 + 160}}{8}
   \]

   \[
   a = \frac{-3 \pm \sqrt{169}}{8}
   \]

   \[
   a = \frac{-3 \pm 13}{8}
   \]

   Calculate the roots:

   \[
   a = \frac{-3 + 13}{8} = \frac{10}{8} = \frac{5}{4}
   \]

   \[
   a = \frac{-3 - 13}{8} = \frac{-16}{8} = -2
   \]

   So, the roots are \(a = \frac{5}{4}\) and \(a = -2\).

4. **Determine the intervals**:

   These roots divide the number line into three intervals: 

   \[
   (-\infty, -2), \quad (-2, \frac{5}{4}), \quad \text{and} \quad \left(\frac{5}{4}, \infty\right)
   \]

   To find where the inequality \(4a^2 + 3a - 10 < 0\) holds, we test a point in each interval:

   - For \((- \infty, -2)\): Test \(a = -3\):

     \[
     4(-3)^2 + 3(-3) - 10 = 36 - 9 - 10 = 17 \quad (\text{positive})
     \]

   - For \((-2, \frac{5}{4})\): Test \(a = 0\):

     \[
     4(0)^2 + 3(0) - 10 = -10 \quad (\text{negative})
     \]

   - For \(\left(\frac{5}{4}, \infty\right)\): Test \(a = 2\):

     \[
     4(2)^2 + 3(2) - 10 = 16 + 6 - 10 = 12 \quad (\text{positive})
     \]

   The inequality is satisfied in the interval where the quadratic expression is negative:

   \[
   (-2, \frac{5}{4})
   \]

Therefore, the solution set in interval notation is:

\[
(-2, \frac{5}{4})
\]

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

1. How do you verify if an inequality is satisfied in a given interval?
2. Can a quadratic inequality have more than two intervals of solution?
3. What is the significance of the sign of the leading coefficient in a quadratic inequality?
4. How do you determine whether to use open or closed intervals in inequalities?
5. What are the possible outcomes when solving quadratic inequalities?

**Tip:** When solving inequalities, always test points from each interval created by the roots to determine where the inequality holds true.

Learn how to solve the quadratic inequality 4a^2 + 8a < 5(a + 2) step-by-step with detailed explanations and solutions. This problem involves quadratic equations and inequalities, suitable for advanced high school students.

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