\(\frac{2x+3}{2x+1}-\frac{2x+5}{2x+7}=1-\frac{6x^2+9x-9}{\left(2x+1\right)\left(2x+7\right)}\)
\(\Leftrightarrow1+\frac{2}{2x+1}-1-\frac{2}{2x+7}-1=-\frac{6x^2+9x-9}{\left(2x+1\right)\left(2x+7\right)}\)
\(\Leftrightarrow\frac{4x+14-4x-2+6x^2+9x-9}{\left(2x+1\right)\left(2x+7\right)}=1\)
\(\Leftrightarrow\frac{6x^2+9x+3}{\left(2x+1\right)\left(2x+7\right)}=1\)
\(\Leftrightarrow\frac{6x^2+6x+3x+3}{\left(2x+1\right)\left(2x+7\right)}=1\)
\(\Leftrightarrow\frac{6x\left(x+1\right)+3\left(x+1\right)}{\left(2x+1\right)\left(2x+7\right)}=1\)
\(\Leftrightarrow\frac{3\left(2x+1\right)\left(x+1\right)}{\left(2x+1\right)\left(2x+7\right)}=1\)
\(\Leftrightarrow3x+3=2x+7\)
\(\Leftrightarrow x=4\)
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