Consider the integral expression $$P = ( x - 1 ) ^ { 2 } ( y + 5 ) + ( 2 x - 3 ) ( y + 4 ) - ( x - 1 ) ^ { 2 } .$$ (1) $P$ can be transformed into $$P = \left( x ^ { 2 } - \mathbf { K } \right) ( y + \mathbf { L } ) .$$ (2) The pairs $( x , y )$ of integers $x$ and $y$ which give $P = 7$ are $$( \pm \mathbf { M } , \mathbf { N O P } ) , \quad ( \pm \mathbf { Q } , \mathbf { R S } ) .$$ (3) Let $a$ be a rational number. If $x = \sqrt { 2 } + 2 \sqrt { 3 }$ and $y = a + \sqrt { 6 }$, then the value of $a$ such that the value of $P$ is a rational number is $\mathbf { T U }$.
Consider the integral expression
$$P = ( x - 1 ) ^ { 2 } ( y + 5 ) + ( 2 x - 3 ) ( y + 4 ) - ( x - 1 ) ^ { 2 } .$$
(1) $P$ can be transformed into
$$P = \left( x ^ { 2 } - \mathbf { K } \right) ( y + \mathbf { L } ) .$$
(2) The pairs $( x , y )$ of integers $x$ and $y$ which give $P = 7$ are
$$( \pm \mathbf { M } , \mathbf { N O P } ) , \quad ( \pm \mathbf { Q } , \mathbf { R S } ) .$$
(3) Let $a$ be a rational number. If $x = \sqrt { 2 } + 2 \sqrt { 3 }$ and $y = a + \sqrt { 6 }$, then the value of $a$ such that the value of $P$ is a rational number is $\mathbf { T U }$.