Left-align math and also align by equal sign?

The fl in flalign and flalign* is short for "full length", not "flush left". Please see page 8 of the user guide of the amsmath package for some use cases for flalign. (Aside: Just because flalign can be abused to force displayed equations to be typeset left-aligned doesn't mean that it's harmless to do so.)

If the objective is to typeset displayed equations left-aligned rather than centered, you really shouldn't be using flalign. Instead, load the amsmath or mathtools package with the option fleqn (short for "flush-left equations") and set the value of the \mathindent length variable to 0pt.

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\usepackage{showframe} % optional

    f(x) &= -1.25x^{2} + 1.5x + 1.66 \\
         &= -1.25x^{2} + 1.5x + 1.66 + 0 % note: no '\\' at end of final line of 'align'

To align at the =, you need to have the & to be just before the =:

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    f(x) &= -1.25x^{2} + 1.5x + 1.66     & \\
         &= -1.25x^{2} + 1.5x + 1.66 + 0 & \\