To solve the quadratic inequality [tex]\(x^2 + 7x + 10 < 0\)[/tex], we'll follow these steps:
1. Find the roots of the quadratic equation [tex]\(x^2 + 7x + 10 = 0\)[/tex].
2. Use the roots to determine the intervals to test.
3. Check each interval to see where the inequality holds true.
### Step 1: Find the Roots of the Quadratic Equation
We start by solving the quadratic equation [tex]\(x^2 + 7x + 10 = 0\)[/tex].
To do this, we can factor the quadratic expression:
[tex]\[ x^2 + 7x + 10 = (x + 5)(x + 2) \][/tex]
Setting each factor equal to zero gives us the roots:
[tex]\[ x + 5 = 0 \quad \Rightarrow \quad x = -5 \][/tex]
[tex]\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \][/tex]
So, the roots of the equation are [tex]\(x = -5\)[/tex] and [tex]\(x = -2\)[/tex].
### Step 2: Determine the Intervals
The roots divide the real number line into three intervals:
1. [tex]\( (-\infty, -5) \)[/tex]
2. [tex]\( (-5, -2) \)[/tex]
3. [tex]\( (-2, \infty) \)[/tex]
### Step 3: Test Each Interval
We need to test each interval to determine where the inequality [tex]\(x^2 + 7x + 10 < 0\)[/tex] holds.
Interval 1: [tex]\( (-\infty, -5) \)[/tex]
Choose a test point in this interval, such as [tex]\(x = -6\)[/tex]:
[tex]\[ (-6)^2 + 7(-6) + 10 = 36 - 42 + 10 = 4 \][/tex]
Since [tex]\(4 \ge 0\)[/tex], the inequality does not hold in this interval.
Interval 2: [tex]\( (-5, -2) \)[/tex]
Choose a test point in this interval, such as [tex]\(x = -3\)[/tex]:
[tex]\[ (-3)^2 + 7(-3) + 10 = 9 - 21 + 10 = -2 \][/tex]
Since [tex]\(-2 < 0\)[/tex], the inequality holds in this interval.
Interval 3: [tex]\( (-2, \infty) \)[/tex]
Choose a test point in this interval, such as [tex]\(x = 0\)[/tex]:
[tex]\[ 0^2 + 7(0) + 10 = 10 \][/tex]
Since [tex]\(10 \ge 0\)[/tex], the inequality does not hold in this interval.
### Conclusion
The inequality [tex]\(x^2 + 7x + 10 < 0\)[/tex] holds in the interval [tex]\((-5, -2)\)[/tex].
Therefore, the solution in interval notation is:
[tex]\[
\boxed{(-5, -2)}
\][/tex]
