How do you solve this differential equation using variation of parameters?

How do you solve this differential equation using variation of parameters?

$\color{green}{question}$:
How do you solve this differential equation using variation of parameters?
$$y"-\frac{2x}{x^2+1}y'+\frac{2}{x^2+1}y=6(x^2+1)$$
$\color{green}{I~tried}$ . . .
$using~the~\color{blue}{Laplace~transform}~method$ . . .
$$L[\int_{0}^{\infty }\frac{sinxt}{1+t^{2}}dt]$$
$$=\int_{0}^{\infty }e^{-sx}(\int_{0}^{\infty }\frac{sinxt}{1+t^{2}}dt)$$
$$=\int_{0}^{\infty }\frac{1}{1+t^{2}}(\int_{0}^{\infty
}e^{-px}sinxtdx)dt\\\\\\=\int_{0}^{\infty
}\frac{1}{1+t^{2}}\frac{t}{s^{2}+t^{2}}dt$$
$$=\int_{0 }^{\infty
}\frac{1}{s^{2}-1}(\frac{t}{1+t^{2}}-\frac{t}{s^{2}+t^{2}})$$
$$=\frac{Lns}{s^{2}-1}$$
Is my solution correct?
Should I use the inverse Laplace?
How can I get a complete and correct answer?
Thanks for any hint.