DISCLAIMER: This question is divided into two parts, the first is a generalization of the second one, and the majority of the mathematics community in StackExchange will answer it and ignore the second, the problem is inspired from chemistry, but I want to know the underlying logic and mathematical tools exploited to come up with the solution
Problem Number 1: let $P(x)$ be a finite degree polynomial such as $$P(x)=\sum_{i=0}^n{a_ix^i}$$ such as $$\forall a_i \neq0$$ let $x$ be a function of time $t$ such as $$\frac{dx}{dt}=P(x)$$ how can one solve this differential equation in the general form.
Problem Number 2 let $A,B,C,D$ be chemical elements, such as $A$ and $B$ are the reactants and $C$ and $D$ are the products. Let their coefficients be $\alpha,\beta,\gamma,\delta$ such as $$\alpha A+\beta B \rightarrow \gamma C+\delta D$$One can express the quantity of each element as $$n_i(t)=n_i(0)-\alpha x(t)$$ with $x(t)$ denoting how far the experiment went on (sorry I am a native french speaker and I would need some help with the translation). Without much talking we can express the speed of our reaction as $$V=-\frac{1}{\alpha_i} \frac{d[A_i]}{dt}=\frac{1}{b_i} \frac{d[B_i]}{dt}$$ with $\alpha_i$ the stoechiometric cofficient of the $i^{th}$ reactant $A_i$, and $\beta_i$ for the $i^{th}$ product $B_i$. In our particular reaction this can be translated into$$V=-\frac{1}{\alpha}\frac{d[A]}{dt}=-\frac{1}{\beta}\frac{d[B]}{dt}=\frac{1}{\gamma}\frac{d[C]}{dt}=\frac{1}{\delta}\frac{d[D]}{dt}$$ Since $n(A)=n_0(A)-\alpha x$ and $n(B)=n_0(B)-\beta x$ $$\implies\frac{\beta}{\alpha}([A]-[A]_0)+[B]_0=[B]$$With $[A]_0$ the initial concentration of $A$ and $[B]_0$ the initial concentration of $B$ (Volume is assumed to be constant). We assume that our reaction is a second order elementary reaction this means that $$V=[A]^m[B]^n$$ with $m$ and $n$ both deduced experimentally. The Above equation implies obviously that $$\frac{d[A]}{dt}=-\alpha[A]^m(\frac{\beta}{\alpha}[A]-\frac{\beta}{\alpha}[A]_0 + [B]_0)^n$$Which can be written differently as $$\frac{d[A]}{dt}=-\alpha \sum_{i=0}^n{(\frac{\beta}{\alpha})^i[A]^{m+i}([B]_0-\frac{\beta}{\alpha}[A]_0)^{n-i}}$$Which is equivalent to say$$\frac{d[A]}{dt}=\sum_{i=0}^n{\alpha_i [A]^{i+k}}$$ So how could we solve second order reaction equations like this? (Computer Algorithms and heuristical approximations are fully accepted as an answer)
As for your first question, any first order nonlinear ordinary differential equation of the form $$\frac{dx}{dt}=P(x)$$ is separable. We can simply write
$$\frac{dx}{P(x)}=dt$$ because we know that $dx = \frac{dx}{dt}dt = P(x) dt .$ Now we may integrate from $t = t_0$ to $t = t_f$ as
$$\int_{x(t_0)}^{x(t_f)}\frac{dx}{P(x)}=\int_{t_0}^{t_f}dt $$
and that is all there is to it.
As for your second question, you may use the same method. If there are many coefficients $\alpha_i $ or if you would simply prefer to use a numerical method, you may run a numerical integration algorithm on
$$ \int_{[A]_0}^{[A]_f} \frac{dA}{\sum_{i=0}^{n} \alpha_i [A]^{i+k} }$$
where $[A_0]$ and $[A_f]$ are the initial and final concentrations of compound $A$ respectively. It would make more sense however to use a finite difference style method like the Euler's Method or RK4 method (just to name a couple). For example, the forward Euler method discretizes a differential equation of the form $x'(t) = f(t,x)$ as
$$\frac{x_{k+1} - x_k}{h} = f(t_k,x_k)$$ for some time step size $h.$ You may then iterate the method as
$$x_{k+1} = x_k + hf(t_k,x_k)$$ where $t_k = t_0 + hk.$
In your case, $f = f(x) = P(x)$ from your first question's formulation so you won't have to provide a $t_0$ and only an $x_0.$ For your second question this you will provide an initial concentration $[A]_0.$
If you need anything else feel free to ask.