Answer:
100°
Step-by-step explanation:
You want the measure of angle MRP, given MN and PQ intersect at R with angle NRP = (2x² -18)° and angle MRQ = (x² +4x +3)°.
Vertical angles
Angles NRP and MRQ are vertical angles, so their measures are the same:
2x² -18 = x² +4x +3
x² -4x -21 = 0 . . . . . . . . subtract the right side expression
(x -7)(x +3) = 0 . . . . . . factor the quadratic
x = 7 or -3 . . . . . . . . values that make the factors zero
Using these values in the angle expressions, we find ...
2x² -18 = 2{7, -3}² -18 = 2{49, 9} -18 = {98, -18} -18 = {80, 0}
We take the zero angle value to be extraneous, so we have ...
∠NRP = ∠MRQ = 80°
Linear pair
The linear pair of angles MRP and NRP are supplementary, so we have ...
∠MRP = 180° -∠NRP
∠MRP = 180° -80°
∠MRP = 100°
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Additional comment
If you consider ∠NRP = 0° to be a legitimate solution, then ∠MRP = 180° is another solution to this problem.
