<s>[INST] Double the value of a quadratic expression If I have this expression $f(x)=ax^2+bx+c$ and I want an expression for $2f(x)$ can I get that with a given change of $x$ ? For example, I have $f(x)=2x^2+3x+9$ and I want to find which factor for $x$ would make $f(x)$ twice the value at every point. Basically I'm looking for another function $g(x)$ so that I can use that and then $f(g(x))=2f(x)$.
 [/INST] Let's try your example, $f(x)=2x^2+3x+9$.
You require that
$$ 2(g(x))^2+3g(x)+9 = f(g(x))= 2f(x)=4x^2+6x+18.$$
As a quadratic equation in $g(x),$ this is
$$ 2(g(x))^2 + 3g(x) - (4x^2+6x+9) = 0$$
and its solution by the quadratic formula is
\begin{align}
g(x) &= \frac{-3 \pm \sqrt{3^2 - 4(2)(- (4x^2+6x+9))}}{2(2)} \\
&= \frac{-3 \pm \sqrt{32x^2 + 48x + 81}}{4} \\
\end{align}
Pick your choice of either $+$ or $-$ for the $\pm$ sign and you have a function,
for example,
$$
g(x) = -\frac34 + \frac14 \sqrt{32x^2 + 48x + 81}.
$$

Note that $f(x)=2x^2+3x+9$ has no real roots.
For a quadratic polynomial with just one real root,
$g(x)$ can be simpler.
For example, if we let $f(x) = x^2 + 4x + 4,$ then
$$ (g(x))^2+4g(x)+4 = f(g(x)) = 2f(x)=2x^2+8x+8, $$
$$ (g(x))^2+4g(x)-(2x^2+8x+4) = 0, $$
\begin{align}
g(x) &= \frac{-4 \pm \sqrt{4^2 - 4(- (2x^2+8x+4))}}{2} \\
&= -2 \pm \frac14 \sqrt{8x^2 + 32x + 32} \\
&= -2 \pm \frac14 \sqrt{8(x^2 + 4x + 4)} \\
&= -2 \pm \frac{\sqrt2}{2} (x + 2) \\
\end{align}
and you can choose a sign from $\pm$ to get a single function such as
$$ g(x) = -2 + \frac{\sqrt2}{2} (x + 2) 
=  \frac{\sqrt2}{2} x + \left({\sqrt2} - 1\right). $$

If the quadratic polynomial has two roots, however, it will have either a minimum value that is negative or a maximum value that is positive.
If there is a minimum value, it is not possible to get twice the minimum value by substituting $g(x)$ for $x$ in the polynomial, because twice the minimum is less than the minimum (which is negative) and by definition there is nothing you can plug into the polynomial that will give you less than the minimum.
For similar reasons, if there is a maximum you cannot get twice the maximum.
The best you can do is to double the polynomial's value for values of $x$ outside some interval around the value where the minimum or maximum of the polynomial occurs.
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