Let $a$ be a constant. Consider the quadratic inequality $$x ^ { 2 } - 2 ( a + 2 ) x + 25 > 0 . \tag{1}$$ The left-hand side of inequality (1) can be transformed into $$( x - a - \mathbf { A } ) ^ { 2 } - a ^ { 2 } - \mathbf { B } a + \mathbf { C D } .$$ Hence, we have the following results. (1) The condition under which inequality (1) holds for all real numbers $x$ is $$\mathbf { E F } < a < \mathbf { G } .$$ (2) The condition under which inequality (1) holds for all real numbers $x$ satisfying $x \geqq - 1$ is $$\mathbf { H I J } < a < \mathbf { K } .$$
Let $a$ be a constant. Consider the quadratic inequality
$$x ^ { 2 } - 2 ( a + 2 ) x + 25 > 0 . \tag{1}$$
The left-hand side of inequality (1) can be transformed into
$$( x - a - \mathbf { A } ) ^ { 2 } - a ^ { 2 } - \mathbf { B } a + \mathbf { C D } .$$
Hence, we have the following results.
(1) The condition under which inequality (1) holds for all real numbers $x$ is
$$\mathbf { E F } < a < \mathbf { G } .$$
(2) The condition under which inequality (1) holds for all real numbers $x$ satisfying $x \geqq - 1$ is
$$\mathbf { H I J } < a < \mathbf { K } .$$