<s>[INST] Sketch curve $y = (4x^3-2x^2+5)/(2x^2+x-3)$ I'm trying to sketch the curve
$$
y = (4x^3-2x^2+5)/(2x^2+x-3).
$$
I tried to find the first and second derivative but I don't know how to find the roots of these.
\begin{align}
y' &= \frac{-5-8 x-38 x^2+8 x^3+8 x^4}{(-3+x+2 x^2)^2}
&&\text{Fermat Theorem}\\
0 &=  -5-8 x-38 x^2+8 x^3+8 x^4
&&\leftarrow\text{Stuck}\\
y'' &= \frac{34+276 x-24 x^2+64 x^3}{(-3+x+2 x^2)^3} \\
0 &= (-5-8 x-38 x^2+8 x^3+8 x^4)(-3+x+2 x^2)
&&\leftarrow\text{Stuck}
\end{align}
I am currently learning Calculus I.
 [/INST] You don't need the second derivative.


*

*Factorise the denominator to find vertical asymptotes.

*Ask yourself what happens when $x \to \pm \infty$ to find the horizontal asymptotes.

*Put $x=0$ to find the $y$-intercept.

*Look for the roots of the numerator to give you the $x$-interecepts.

*Solve $y'=0$ to give yourself the turning points.


Once you have all of these the sketch should be straightforward.
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