The equation of the given curve is y = x3 + 2x + 6. The slope of the tangent to the given curve at any point (x, y) is given by,
If the normal is parallel to the line, then we must have the slope of the normal being equal to the slope of the line
When x = 2, y = 8 + 4 + 6 = 18.
When x = −2, y = − 8 − 4 + 6 = −6.
Therefore, there are two normals to the given curve with slope and passing through the points (2, 18) and (−2, −6).
Thus, the equation of the normal through (2, 18) is given by,
And, the equation of the normal through (−2, −6) is given by,
Hence, the equations of the normals to the given curve (which are parallel to the given line) are 
