CAT 2023 Slot 1 Solved Paper

Let us consider 3 cases:
1) $\mathit{x}=0$, This is a solution, as both L.H.S and R.H.S will be equal ( 0 ) when $\mathit{x}=0$. ( 1 solution)
2) $\mathrm{x}>0$
$\Rightarrow 2\mathit{x}\left({\mathit{x}}^2+1\right)=5{\mathit{x}}^2$
$\Rightarrow 2\left({\mathit{x}}^2+1\right)=5\mathit{x}$
$\Rightarrow 2{\mathit{x}}^2-5\mathit{x}+2=0\Rightarrow 2{\mathit{x}}^2-4\mathit{x}-\mathit{x}-2=0$
$\Rightarrow 2\mathit{x}\left(\mathit{x}-2\right)-1\left(\mathit{x}-2\right)=0$
$\Rightarrow \left(\mathit{x}-2\right)\left(2\mathit{x}-1\right)=0\Rightarrow \mathrm{x}=2$ or $1∕2\Rightarrow$(1 integer solution )
3)
$\Rightarrow -2\mathit{x}\left({\mathit{x}}^2+1\right)=5{\mathit{x}}^2$
$\Rightarrow 2{\mathit{x}}^2+5\mathit{x}+2=0$
$\Rightarrow 2{\mathit{x}}^2+4\mathit{x}+\mathit{x}+2=0$
$\Rightarrow 2\mathit{x}\left(\mathit{x}+2\right)+1\left(\mathit{x}+2\right)=0$
$\Rightarrow \left(\mathit{x}+2\right)\left(2\mathit{x}+1\right)=0\Rightarrow \mathrm{x}=-2$ or $-1∕2\Rightarrow$ (1 integer solution )
So, the total number of integer solutions are $0,2,-2\Rightarrow 3$.