+5 In the diagram, quadrilateral ABCD is inscribed in the circle, arc ADB is a minor arc, and segment AB is parallel to segment DC. Given that arc DC is 30 degrees, arc AD is x^2 + 7x degrees, and arc BC is 60 - 4x degrees, find the measure of arc AEB.
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\([asy] unitsize(2 cm); pair A, B, C, D, E; A = dir(170); B = dir(30); D = dir(130); C = dir(70); E = dir(280); draw(Circle((0,0),1)); draw(A--B--C--D--cycle); label("$A$", A, W); label("$B$", B, dir(0)); label("$C$", C, NE); label("$D$", D, NW); dot("$E$", E, S); [/asy]\)
+130081 Since AB is parallel to BC, then chords AD and BC are equal....thus
x^2 + 7x = 60 - 4x rearrange as
x^2 + 11x - 60 = 0 factor
(x - 4) ( x + 15) = 0 setting each factor to 0 and solving for x we have that
x = 4 or x = -15
Taking the positive solution, then BC = AB = 60 - 4(4) = 60 - 16 = 44
Thus arcs AD + DC + CB = [ 44 + 30 + 44 ] = 118°
Thus arc AEB = 360 - 118 = 242°
