Let a and p be the roots of equation `x^2 + x - 3 = 0`. Find the value of the expression `4p^2 - a^3`.

Answer

To solve the problem, we need to find the roots of the quadratic equation \(x^2 + x - 3 = 0\) and then evaluate the expression \(4p^2 - a^3\), where \(a\) and \(p\) are the roots of the equation.

### Step 1: Find the roots of the equation
The roots of the quadratic equation \(ax^2 + bx + c = 0\) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation \(x^2 + x - 3 = 0\), we have:
- \(a = 1\)
- \(b = 1\)
- \(c = -3\)

Substituting these values into the quadratic formula:
\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1} \]
\[ x = \frac{-1 \pm \sqrt{1 + 12}}{2} \]
\[ x = \frac{-1 \pm \sqrt{13}}{2} \]

Thus, the roots are:
\[ a = \frac{-1 + \sqrt{13}}{2}, \quad p = \frac{-1 - \sqrt{13}}{2} \]

### Step 2: Calculate \(4p^2\)
Now we need to calculate \(4p^2\):
\[ p = \frac{-1 - \sqrt{13}}{2} \]
Calculating \(p^2\):
\[ p^2 = \left(\frac{-1 - \sqrt{13}}{2}\right)^2 = \frac{(-1 - \sqrt{13})^2}{4} \]
Expanding the square:
\[ (-1 - \sqrt{13})^2 = 1 + 2\sqrt{13} + 13 = 14 + 2\sqrt{13} \]
Thus,
\[ p^2 = \frac{14 + 2\sqrt{13}}{4} = \frac{7 + \sqrt{13}}{2} \]
Now, calculate \(4p^2\):
\[ 4p^2 = 4 \cdot \frac{7 + \sqrt{13}}{2} = 2(7 + \sqrt{13}) = 14 + 2\sqrt{13} \]

### Step 3: Calculate \(a^3\)
Next, we need to calculate \(a^3\):
\[ a = \frac{-1 + \sqrt{13}}{2} \]
Calculating \(a^3\):
\[ a^3 = \left(\frac{-1 + \sqrt{13}}{2}\right)^3 = \frac{(-1 + \sqrt{13})^3}{8} \]
Expanding the cube using the binomial theorem:
\[ (-1 + \sqrt{13})^3 = -1 + 3(-1)^2(\sqrt{13}) + 3(-1)(\sqrt{13})^2 + (\sqrt{13})^3 \]
Calculating each term:
- \(3(-1)^2(\sqrt{13}) = 3\sqrt{13}\)
- \(3(-1)(\sqrt{13})^2 = -3 \cdot 13 = -39\)
- \((\sqrt{13})^3 = 13\sqrt{13}\)

Combining these:
\[ (-1 + \sqrt{13})^3 = -1 + 3\sqrt{13} - 39 + 13\sqrt{13} = -40 + 16\sqrt{13} \]
Thus,
\[ a^3 = \frac{-40 + 16\sqrt{13}}{8} = -5 + 2\sqrt{13} \]

### Step 4: Calculate \(4p^2 - a^3\)
Now we can substitute \(4p^2\) and \(a^3\) into the expression:
\[ 4p^2 - a^3 = (14 + 2\sqrt{13}) - (-5 + 2\sqrt{13}) \]
Simplifying this:
\[ 4p^2 - a^3 = 14 + 2\sqrt{13} + 5 - 2\sqrt{13} = 19 \]

### Final Answer
The value of the expression \(4p^2 - a^3\) is:
\[ \boxed{19} \]