To solve the inequality [tex]\( x^3 + 4x^2 > x + 4 \)[/tex], we follow these steps:
1. Move all terms to one side of the inequality:
Subtract [tex]\(x + 4\)[/tex] from both sides to get a single expression on one side:
[tex]\[
x^3 + 4x^2 - x - 4 > 0
\][/tex]
2. Factor the polynomial if possible:
The polynomial [tex]\(x^3 + 4x^2 - x - 4\)[/tex] is not easily factorable, so we need to use other methods such as finding roots through polynomial solving techniques or graphing.
3. Determine the critical points:
Find the roots of the equation [tex]\( x^3 + 4x^2 - x - 4 = 0 \)[/tex]. These roots are points where the polynomial changes sign. Solving for these roots:
Solving [tex]\( x^3 + 4x^2 - x - 4 = 0 \)[/tex] gives us:
[tex]\[
x = -4, x = -1, \text{ and } x = 1
\][/tex]
4. Plot these critical points on a number line:
These points split the number line into intervals. The intervals to test are:
- [tex]\( (-\infty, -4) \)[/tex]
- [tex]\( (-4, -1) \)[/tex]
- [tex]\( (-1, 1) \)[/tex]
- [tex]\( (1, \infty) \)[/tex]
5. Test each interval:
We pick test points from each interval to determine the sign of the polynomial in those intervals.
For interval [tex]\( (-\infty, -4) \)[/tex]:
- Choose [tex]\( x = -5 \)[/tex], [tex]\( x^3 + 4x^2 - x - 4 = -125 + 100 + 5 - 4 = -24 \)[/tex] (negative)
For interval [tex]\( (-4, -1) \)[/tex]:
- Choose [tex]\( x = -2 \)[/tex], [tex]\( x^3 + 4x^2 - x - 4 = -8 + 16 + 2 - 4 = 6 \)[/tex] (positive)
For interval [tex]\( (-1, 1) \)[/tex]:
- Choose [tex]\( x = 0 \)[/tex], [tex]\( x^3 + 4x^2 - x - 4 = 0 + 0 - 0 - 4 = -4 \)[/tex] (negative)
For interval [tex]\( (1, \infty) \)[/tex]:
- Choose [tex]\( x = 2 \)[/tex], [tex]\( x^3 + 4x^2 - x - 4 = 8 + 16 - 2 - 4 = 18 \)[/tex] (positive)
6. Combine the intervals where the polynomial is positive:
Based on our testing:
- [tex]\( x \in (-4, -1) \)[/tex]
- [tex]\( x \in (1, \infty) \)[/tex]
These are the intervals where [tex]\(x^3 + 4x^2 - x - 4 > 0\)[/tex].
Thus, the solution to the inequality [tex]\( x^3 + 4x^2 > x + 4 \)[/tex] is:
[tex]\[
-4 < x < -1 \quad \text{or} \quad x > 1
\][/tex]
Therefore, the correct answer is:
[tex]\[
-4 < x < -1 \quad \text{or} \quad x > 1
\][/tex]
