How to estimate two unknown parameters of a equation

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The equation is $$ y(x) = ab^2x^2 + abx + g(x) $$

I know $x$, $y(x)$ and $g(x)$, I want to estimate $a$ and $b$.

And I want to optimize $a$ and $b$ with future data of $x$ and $y(x)$.

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A procedure for fitting is shown below :

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