To solve the equation \(x^2 = -13x - 4\), we can first rearrange it into a standard quadratic equation format:
\[ x^2 + 13x + 4 = 0 \]
Now, we can use the quadratic formula to find the solutions, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, the coefficients are:
- \(a = 1\)
- \(b = 13\)
- \(c = 4\)
Calculating the discriminant \(b^2 - 4ac\):
\[ b^2 - 4ac = 13^2 - 4 \times 1 \times 4 = 169 - 16 = 153 \]
Now we can plug this into the quadratic formula:
\[ x = \frac{-13 \pm \sqrt{153}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{-13 \pm \sqrt{153}}{2} \]
Thus, the two solutions are:
\[ x = \frac{-13 - \sqrt{153}}{2} \quad \text{and} \quad x = \frac{-13 + \sqrt{153}}{2} \]
So the correct answer is:
\[ \frac{-13 - \sqrt{153}}{2}, \frac{-13 + \sqrt{153}}{2} \]
This matches the last option you provided.