To find the roots of the polynomial \( p(x) = x^3 + 4x^2 - 7x - 10 \), we can apply the Rational Root Theorem and synthetic division to identify rational roots, followed by solving for the remaining roots.
Testing possible rational roots, we can try \( x = -2 \):
\[ p(-2) = (-2)^3 + 4(-2)^2 - 7(-2) - 10 \] \[ = -8 + 16 + 14 - 10 \] \[ = 12 \quad (\text{not a root}) \]
Next, let's try \( x = -1 \):
\[ p(-1) = (-1)^3 + 4(-1)^2 - 7(-1) - 10 \] \[ = -1 + 4 + 7 - 10 \] \[ = 0 \quad (\text{is a root}) \]
Since \( x = -1 \) is a root, we can perform synthetic division of \( p(x) \) by \( x + 1 \):
\[ \begin{array}{r|rrrr} -1 & 1 & 4 & -7 & -10 \ & & -1 & -3 & 10 \ \hline & 1 & 3 & -10 & 0 \ \end{array} \]
The result is \( x^2 + 3x - 10 \).
Next, we need to factor \( x^2 + 3x - 10 \). To find the roots of this quadratic equation, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = 3 \), and \( c = -10 \):
\[ x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-10)}}{2(1)} \] \[ = \frac{-3 \pm \sqrt{9 + 40}}{2} \] \[ = \frac{-3 \pm \sqrt{49}}{2} \] \[ = \frac{-3 \pm 7}{2} \]
Calculating the two roots:
- \( x = \frac{-3 + 7}{2} = \frac{4}{2} = 2 \)
- \( x = \frac{-3 - 7}{2} = \frac{-10}{2} = -5 \)
Thus, the roots of the original polynomial are \( x = -5, -1, 2 \).
Arranging these in ascending order, we have:
\[ \text{The roots are: } -5, -1, \text{ and } 2. \]
So, the final answer is:
The roots are \(-5\), \(-1\), and \(2\).