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If we multiply both sides by x+4,we get $x^2+2x+3=x^2+9x+20$ which can be simplified to $0=7x+17$ubtract 17 from both sides and divide by 7. we get -17/7.

Cross-multiplication gives 

$x^2+2x+3=(x+4)(x+5)=x^2+9x+20.$

Therefore

$0=7x+17$

and $x=\boxed{-\frac{17}7}$.