 Hello friends, and how are you all doing today? My name is Priyanka. The question says, find the equation of the normal to the curve y is equal to xq plus 2x plus 6 which are parallel to the line x plus 14 y plus 4 is equal to 0. My hair, we are given the equation of the curve y as xq plus 2x plus 6. Now, I am differentiating it with respect to x we have dy by dx equal to 3x square plus 2 and therefore this is the slope of the tangent 2 at point x and y. Now, slope of the normal is minus 1 upon slope of the tangent which is equal to minus 1 upon 3x square plus 2. Now, the slope of the given line that is x plus 14 y plus 4 is equal to 0 that means 14 y is equal to minus x minus 4 or y is equal to minus x by 14 minus 4 by 14. So, slope is minus 1 by 14. Now, since we are given in the question that normal is parallel to the line. So, we can say that the slope of the normal that we have found out above is equal to the slope of the given line that is minus 1 by 14. On simplifying we can write it as 3x square plus 2 is equal to 14 that is on reversing the equation which further gives us 3x square minus 12 is equal to 0 or x square is equal to 12 by 3 that is x is equal to plus minus 2. Now, we need to find both the points when x is equal to 2 we need to find the value of y and when x is equal to minus 2 then also we need to find the value of y. The value of y will be 2 cube plus 2 into 2 plus 6 which is equal to 8 plus 4 plus 6 which is equal to 80 and here it will be minus 2 the whole cube plus 2 into minus 2 plus 6 which is minus 8 minus 4 plus 6 which is equal to minus 12 plus 6 which is minus 6. So, the equation of the tangent that passes through 2 points are 2 comma 18 and minus 2 comma minus 6. Now, let us find out the equation of normal points 2 comma 18 and minus 2 comma minus 6 respectively which will be y minus y 1 equal to now slope of the normal found out above was minus 1 by 14 x minus x 1 which implies 14 y minus 18 into 14 gives us 252 equal to minus x plus 2 which gives us x plus 14 y minus 254 is equal to 0. And here we have y plus 6 equal to minus 1 by 14 x plus 2 we have 14 y plus 84 equal to minus x minus 2 which is equal to x plus 14 y plus 86 is equal to 0. So, the answer to the given question is x plus 14 y minus 254 is equal to 0 and x plus 14 y plus 86 is equal to 0 right this completes my solution hope you understood the whole concept well do take care of your calculations and meet you next by the now.