No idea what the cited section's method is, but this ODE is linear:
[tex]\dfrac{\mathrm dy}{\mathrm dx}=4+y-4x+5[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}-y=9-4x[/tex]
Multiply both sides by [tex]e^{-x}[/tex] so that the left side can be condensed as the derivative of a product:
[tex]e^{-x}\dfrac{\mathrm dy}{\mathrm dx}-e^{-x}y=(9-4x)e^{-x}[/tex]
[tex]\dfrac{\mathrm d}{\mathrm dx}\left[e^{-x}y\right]=(9-4x)e^{-x}[/tex]
Integrating both sides gives
[tex]e^{-x}y=(4x-5)e^{-x}+C[/tex]
[tex]\implies\boxed{y(x)=4x-5+Ce^x}[/tex]
