The number of solutions $x$ of the equation $\sin \left(x+x^2\right)-\sin \left(x^2\right)=\sin x$ in the interval $\left[2,3\right]$ is

Given,

$\sin \left(x+x^2\right)-\sin \left(x^2\right)=\sin x$

$\Rightarrow \sin \left(x+x^2\right)=\sin \left(x^2\right)+\sin x$

$\Rightarrow 2\sin \left(\frac{x+x^2}{2}\right)\cos \left(\frac{x+x^2}{2}\right)=2\sin \left(\frac{x^2+x}{2}\right)\cos \left(\frac{x^2-x}{2}\right)$

$\sin \left(\frac{x+x^2}{2}\right)=0$

or $\cos \left(\frac{x^2+x}{2}\right)-\cos \left(\frac{x^2-x}{2}\right)=0$

$\frac{x^2+x}{2}=x\pi$ or $2\sin \frac{x^2}{2}\sin \frac{x}{2}=0$

$\Rightarrow \frac{x^2+x}{2}=0,\pi ,2\pi$ or $\frac{x^2}{2}=0,\pi ,\frac{n}{2}=0,\pi ,2\pi$

$\Rightarrow x^2+x=2\pi$ or $x^2=2\pi$

$\Rightarrow x^2+x-2\pi =0$ or $x=\sqrt{2\pi }$

$\Rightarrow x=\frac{-1\pm \sqrt{1+8\pi }}{2}$ or $2<\sqrt{2}\pi <3$

$\Rightarrow \sqrt{1+8\pi }\approx \sqrt{25.14}$

$\therefore x=\frac{5.2-1}{2}=\frac{42}{2}=2.1$

$\because$ Total numbers of solution lies between $\left(2,3\right)=2$