At what point does the normal to y=2+1x+4x2 at (1,7) intersect the parabola a second time

At what point does the normal to y=2+1x+4x2 at (1,7) intersect the parabola a second time?

( __________, ___________ )

(The normal line is perpendicular to the tangent line. If two lines are perpendicular their slopes are negative reciprocals -- i.e. if the slope of the first line is m then the slope of the second line is −1/m)

y = 4x^2 + x + 2

y ' = 8x + 1 <===

y ' (1) = 8*1 + 1 = 9 <=== slope of tangent

. . . slope of normal = - 1/9

y = mx + b
m = -1/9 ; x = 1 ; y = 7

7 = -1/9 * 1 + b
b = 64/9

y = -1/9 x + 64/9 <=== normal

normal intersects where

-1/9 x + 64/9 = 4x^2 + x + 2

4 x^2 + 10 / 9 x - 46/9 = 0

2/9 (x - 1) (18x + 23) = 0

x = 1 ... or ...
x = - 23/18 <===

y = -1/9 * - 23/18 + 64/9 = 1175 / 162

(-23/18 , 1175/162) <=== second intersection point.