Let the common tangent line be y = mx+b. Then (x^4 - 4x^2 - 3x) - (mx + b) has two double roots.
(x^2 - 2)^2 completes the square for the higher exponents of x^4 - 4x^2 - 3x. Then (x^2 - 2)^2 = (x^4 - 4x^2 - 3x) - (mx + b).
Expanding and taking linear and constant coefficients implies 0 = -3 - m and 4 = -b. Therefore the common tangent line is y = -3x - 4.
The roots of (x^2 - 2)^2 indicate the x coordinates of the tangent points, x=+/-sqrt(2). Then the tangent points are (sqrt(2), -4 - 3*sqrt(2)) and (-sqrt(2), -4 + 3*sqrt(2))
Edited on July 25, 2019, 10:43 am
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