Answer: 105 degrees
Explanation: In a quadrilateral that is inscribed in a circle, the opposite angles are supplementary. Since angle A and angle C are opposite angles, they are supplementary. In terms of equation,
[tex]m \angle A + m \angle C = 180^{\circ}
\\ \indent (x^2+50)^{\circ} + (12x+45)^{\circ} = 180^{\circ}
\\ \indent (x^2 + 12x + 95)^{\circ} = 180^{\circ}
\\ \indent x^2 + 12x + 95 = 180
\\ \indent x^2 + 12x - 85 = 0
\\ \indent (x - 5)(x + 17) = 0
\\ \indent \boxed{x = 5 \text{ or } x = -17}[/tex]
Note that if x = -17,
[tex]m \angle C = 12x+45
\\ \indent = 12(-17) + 45
\\ \indent m \angle C = -159[/tex]
which is not valid because angle measure is not negative.
So, x = 5. Hence,
[tex]m \angle C = (12x+45)^{\circ}
\\ \indent = (12(5) +45)^{\circ}
\\ \indent \boxed{m \angle C = 105^{\circ}}[/tex]
