Solving for m at an arbitrary value of x would give you the tangent at some point. What you're after though is the rate of change in that slope as x approaches the normal, otherwise known as finding a gradient. Admittedly, this technique would not work in most cases without utilizing limits, as the error in calculation would be fairly substantial. So, really, what the question is reduced to is how to find a tangent without calculus.
Yes, considering tangents is the right way to do this problem.
I would set the formula for slope equal to the quadratic,

y-y_1=m(x-x_1), (equation for slope-intercept)

would then be,

m(x-x_1)+y_1 = ax^2+bx+c, setting the discriminant equal to zero and factoring to obtain a real solution.
You're actually very close to an exact method, if you just expand on this a little.
Also, power series has been around since Babylonian times, much before calculus. You're probably thinking about Taylor or Maclaurin series.
Power series are infinite series, and you tend to find them by Taylor's method. Considering the Babylonians had only just invented 0, I find it hard to believe that they could conceptualise or even come close to requiring as advanced concepts as these. What do you understand by the term 'power series', Mr. Feynman?