Let's first simplify the equation given:
\[ x^2 + 5x^2 + 10x + 50 = 0 \]
Combining like terms, we have:
\[ 6x^2 + 10x + 50 = 0 \]
Next, we can apply the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 6 \), \( b = 10 \), and \( c = 50 \).
First, we need to calculate the discriminant (\( b^2 - 4ac \)):
\[ b^2 = 10^2 = 100 \] \[ 4ac = 4 \cdot 6 \cdot 50 = 1200 \] \[ b^2 - 4ac = 100 - 1200 = -1100 \]
Since the discriminant is negative (\( -1100 < 0 \)), this implies that the quadratic equation has no real solutions, only non-real (imaginary) solutions.
Given that a quadratic equation can have either two real solutions or two non-real (complex) solutions, the solution set in this case consists of two non-real (imaginary) solutions.
Therefore, the correct description of the solution set from the provided options is:
A: It has one real solution and two non-real solutions.
However, it should be noted that the wording could be interpreted to mean that there are no real solutions at all based on the calculations conducted since there are two non-real solutions.
Since the best fit from the options given is not entirely accurate (i.e., there are actually two non-real solutions, not "one real and two non-real"), the correct potential review of the options could point to a lack of accurate reflection in the options. Based on the problem's restriction, it emphasizes the presence of imaginary solutions, thereby leading to a clarifying skepticism towards the mix of real/imaginary counts in the option set.
In conclusion, since Option A does not accurately describe the solutions (there are no real solutions), the best answer is:
E: We cannot tell from the information provided due to inconsistencies in options about real or imaginary solutions.
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