To demonstrate that the Fundamental Theorem of Algebra is true for the quadratic polynomial \(-4x^2 - 24x - 36 = 0\), we can solve the equation using the quadratic formula. The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the quadratic polynomial \(-4x^2 - 24x - 36\), we identify the coefficients as follows:
- \(a = -4\)
- \(b = -24\)
- \(c = -36\)
Next, we will calculate the discriminant, \(D = b^2 - 4ac\):
\[ D = (-24)^2 - 4(-4)(-36) = 576 - 576 = 0 \]
Since the discriminant \(D\) is equal to 0, there is exactly one unique solution (a repeated root) for the quadratic equation. Now we can plug in the values into the quadratic formula:
\[ x = \frac{-(-24) \pm \sqrt{0}}{2 \cdot -4} = \frac{24 \pm 0}{-8} = \frac{24}{-8} = -3 \]
Therefore, the only solution to the equation is \(x = -3\).
The solution set can be accurately described as "two identical solutions" because the root is repeated due to the discriminant being 0.
So, the correct choice is:
There are two identical solutions.