To solve the inequality [tex]\(x^3 + 4x^2 > x + 4\)[/tex], we can follow these steps:
1. Rewrite the Inequality: First, bring all terms to one side of the inequality to facilitate analysis. This gives us:
[tex]\[
x^3 + 4x^2 - x - 4 > 0
\][/tex]
2. Find the Roots of the Equation: We need to find the roots of the corresponding equation:
[tex]\[
x^3 + 4x^2 - x - 4 = 0
\][/tex]
The roots are important because they are the points where the cubic polynomial changes sign.
3. Determine the Intervals: Once we have the roots, we can determine the intervals where the polynomial is positive or negative by testing points within each interval.
For a cubic polynomial [tex]\( ax^3 + bx^2 + cx + d \)[/tex], it will typically change signs at the roots or critical points.
4. Combine the Intervals: We combine the intervals where the polynomial is positive to get the solution to the inequality.
Upon solving, we find that the roots of [tex]\( x^3 + 4x^2 - x - 4 = 0 \)[/tex] are approximately [tex]\( x = -4 \)[/tex], [tex]\( x = -1 \)[/tex], and [tex]\( x = 1 \)[/tex].
By testing points in the intervals defined by these roots:
- For [tex]\(x < -4\)[/tex], the expression [tex]\(x^3 + 4x^2 - x - 4\)[/tex] is negative.
- For [tex]\(-4 < x < -1\)[/tex], the expression is positive.
- For [tex]\(-1 < x < 1\)[/tex], the expression is negative.
- For [tex]\(x > 1\)[/tex], the expression is positive.
Therefore, the polynomial [tex]\(x^3 + 4x^2 - x - 4\)[/tex] is greater than zero in the following intervals:
- [tex]\(-4 < x < -1\)[/tex]
- [tex]\(x > 1\)[/tex]
Thus, combining these positive intervals, 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]
Hence, the correct answer is:
[tex]\[
-4 < x < -1 \quad \text{or} \quad x > 1
\][/tex]
