You can use Variation of Parameters.
The auxiliary equation is \(\displaystyle \L\\m(m-1)+4m-10=(m-2)(m+5)=0\)
Thus, \(\displaystyle \L\\y_{c}=C_{1}x^{2}+C_{2}x^{-5}\)
Divide the equation by x^2 and get: \(\displaystyle \L\\y''+\frac{4}{x}y'-\frac{10}{x^{2}}y=\frac{3}{x^{7}}\)
Make the identification \(\displaystyle \L\\f(x)=\frac{3}{x^{7}}\).
Now with \(\displaystyle \L\\y_{1}=x^{2} \;\ and \;\ y_{2}=x^{-5}\), build the Wronskians:
\(\displaystyle \L\\W, \;\ W_{1}, \;\ W_{2}, \;\ u'_{1}, \;\ u'_{2}\)
Write back if you remain stuck.