15 Point $A$ lies on the curve $y=x^{2}+5 x+8$
The $x$-coordinate of $A$ is -4
15 (a) Show that the equation of the normal to the curve at $A$ is $3 y=x+16$

Answer

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Answer

Box the final answer: \boxed{3y = x + 16}

Steps

Step 1 ：Find the coordinates of point A: (-4, 4)

Step 2 ：Find the derivative of the curve equation: y' = 2x + 5

Step 3 ：Find the slope of the tangent at point A: -3

Step 4 ：Find the slope of the normal: 1/3

Step 5 ：Use the point-slope form to find the equation of the normal: y - 4 = 1/3(x + 4)

Step 6 ：Simplify the equation: 3y = x + 16

Step 7 ：Check the equation meets the requirements

Step 8 ：Box the final answer: \boxed{3y = x + 16}