Calculus Allowed (But It Won't Be Pretty)

Let $m = \sqrt{x^{4} -3x^{2}+ 6x+ 13} - \sqrt{x^{4} -x^{2}+ 1}$

$m = \sqrt{(x^{4} -4x^{2} + 4) + (x^{2} + 6x + 9)} - \sqrt{(x^{4} -2x^{2} + 1) + x^2}$

$m = \sqrt{(x^{2} -2)^2 + (x + 3)^2} - \sqrt{(x^{2} -1)^2 + x^2}$

Now, let $y = x^2$ $m = \sqrt{(y-2)^2 + (x + 3)^2} - \sqrt{(y-1)^2 + x^2}$

So, with reference to figure, it becomes, $m = AP - BP$

Now, using triangle inequality, $AP - BP < AB$

So, $AP - BP_{max} = AB$ (when APB is a straight line)

$m_{max} = AB$

$m_{max} = \sqrt{(0+3)^2 + (1-2)^2}$

$m_{max} = \sqrt{10}$

$α = 10$