The number of points in (- ∞ , ∞) , for which x ^ 2 - xsinx - x = 0...

$f\left(x\right) = {x}^{2} - xsinx - \cos x$

${f}^{'}\left(x\right) = 2x - xcosx - \sin x + \sin x$

 $= x\left(2 - \cos x\right)$

We know that, \left(2 - \cos x\right)$is positive for all real numbers, now it depends on$x$. </p><p>So,$f\left(x\right)is increasing if $x > 0$

$f\left(x\right)$ is decreasing if $x < 0$

$f\left(0\right) = - 1$

$f\left(\infty\right) = \infty$

$f\left(- \infty\right) = \infty$

So, it is clear that graph cuts the $x$-axis at two points, so number of solutions are two.