Question

How do you find the slope of the tangent line to the curve 2x^2+4xy+2y^3=114 at the point (-7,4) ??

Answer 1

Do you know how to do "implicit differentiation" ?
For a refresh, see
https://www.khanacademy.org/math/differential-calculus/taking-derivatives/implicit_differentiation/v/implicit-differentiation-1

you will want to find dy/dx (as a function of x and y) using implicit differentiation.
Evaluate at x=-7, y=4 to find the slope of the tangent line (i.e. the line you are looking for)
If we call the slope m= dy/dx at (-7,4)
find the equation of the line using
\[ y - y_0 = m(x-x_0) \]
where \( (x_0,y_0) = (-7,4) \)