Equations of the Ellipse and Parabola:
The given equations are:
Ellipse: x² + 4y² = 16
Parabola: y² - 4x - 4 = 0

Finding the Slopes of Tangent Lines:
To find the slopes of the tangent lines to the parabola and ellipse, we need to differentiate their respective equations.

Differentiating the Ellipse Equation:
Differentiating the equation of the ellipse with respect to x, we get:
2x + 8y(dy/dx) = 0
Simplifying the equation, we have:
dy/dx = -x / (4y)

Differentiating the Parabola Equation:
Differentiating the equation of the parabola with respect to x, we get:
2yy' - 4 = 0
Simplifying the equation, we have:
y' = 2 / y

Finding the Slopes of the Tangent Lines:
To find the slopes of the tangent lines, we substitute the values of y from the parabola equation into the ellipse equation, and vice versa.

Substituting y from the Parabola into the Ellipse:
Substituting y' = 2 / y into the ellipse equation, we get:
x² + 4(1/4y)(2/y)² = 16
Simplifying the equation, we have:
x² + 8/y² = 16
Rearranging the equation, we get:
x² = 16 - 8/y²
x² = (16y² - 8) / y²
Cross-multiplying, we get:
x²y² = 16y² - 8
x²y² - 16y² = -8
y²(x² - 16) = -8
y² = -8 / (x² - 16)

Substituting x from the Ellipse into the Parabola:
Substituting x² = 16 - 8/y² into the parabola equation, we get:
y² - 4(16 - 8/y²) - 4 = 0
Simplifying the equation, we have:
y² - 64 + 32/y² - 4 = 0
y² - 32 + 32/y² = 0
Multiplying through by y², we get:
y⁴ - 32y² + 32 = 0
This is a quadratic equation in y². Let's call it equation A.

Quadratic Equation with Roots as Tangent Slopes:
The quadratic equation with roots as the slopes of the tangent lines to the parabola and ellipse is equation A: y⁴ - 32y² + 32 = 0.