Quadrilateral ABCD is inscribed in a circle such that m∠A=(x2+78)∘ and m∠C=(10x+46)∘

This is known as a cyclic quadrilateral......the opposite angles are suppllementary

So

x^2 + 78  + 10x + 46  = 180

x^2 + 10x + 124  =  180       subtract 180 from both sides

x^2 + 10x - 56  =  0  factor

(x - 4) ( x + 14)  = 0

Setting each factor to 0  and solving for x  produces

x = 4       and   x  =  -14

Reject the second since it produces negative angles

So  C  =   10(4)  + 46   =  86°