Regard y as independent variable and x as dependant variable and find the slope of the tangent line to the curve (4x^2 + 2y2)^2 - x^2y = 4588 at point (3,4).

Correct answer is -0.668827160493827

Here's what I did:

2(8x(dy/dx) + 4y) -2x(dx/dy)y + x^2 = 0
16x(dx/dy) + 8y - 2x(dx/dy)y = -x^2
16x(dx/dy) - 2x(dx/dy)y = -x^2 - 8y
(dx/dy)(16x -2xy) = -x^2 - 8y
(dx/dy) = (-x^2 - 8y)/(16x -2xy)

But this doesnt work.

You have
(4x2 + 2y2)2 - x2y = 4588
It appears you started correctly and you know what you want, dy/dx, but I don't think you did the correct calculations.
It might help to think of this as
(f(x) + g(y))2 - h(x)*y = C
where f(x) = 4x2, g(y)=2y2 and h(x)=x2.
Then the differentiation should go
2(f(x) + g(y))*d/dx(f(x) + g(y)) - h(x)*dy/dx -h'(x)y = 0
Then f'(x)=8x, d/dx g(y)=g'(y)*dy/dx and h'(x)=2x
What I suggest, and this what I've done for real long expressions, is to write the symbols out, do the differentiation on the symbols, then substitute them into your equation.
Does this help?

Yes, that approach will work. Let's go through the steps together.

First, we have the equation:

(4x^2 + 2y^2)^2 - x^2y = 4588

To find the slope of the tangent line at a given point (3,4), we need to differentiate the equation implicitly with respect to x.

Let's break down the equation into its components:

f(x) = 4x^2
g(y) = 2y^2
h(x) = x^2

Now, let's differentiate each component with respect to the corresponding variable:

df(x)/dx = 8x
dg(y)/dy = 4y
dh(x)/dx = 2x

Now, we'll substitute these derivatives back into the differentiation equation:

2(f(x) + g(y))*df(x)/dx - h(x)*dy/dx - h'(x)*y = 0

Putting the derivatives in the equation, we have:

2((4x^2 + 2y^2))*8x - (x^2)*dy/dx - (2x)*y = 0

Simplifying further:

16x(4x^2 + 2y^2) - x^2*dy/dx - 2xy = 0

Now we substitute the given point (3,4) into the equation:

16(3)(4(3)^2 + 2(4)^2) - (3)^2*dy/dx - 2(3)(4) = 0

Evaluating this equation will give us the value of dy/dx, which is the slope of the tangent line at the point (3,4).