\[(x + 1)(x + 7) - (x + 2)(x + 3) =\]
\[= x^{2} + 7x + x + 7 - \left( x^{2} + 3x + 2x + 6 \right) =\]
\[= x^{2} + 8x + 7 - x^{2} - 3x - 2x - 6 =\]
\[= 3x + 1.\]