Q1 Let $a$ and $b$ be rational numbers and let $p$ be a real number. Consider the quadratic equation

$$x ^ { 2 } + a x + b = 0 \tag{1}$$

which has a solution $x = \frac { \sqrt { 5 } + 3 } { \sqrt { 5 } + 2 }$, and consider the inequality

$$x + 1 < 2 x + p + 3 . \tag{2}$$

(1) First, we are to find the values of $a$ and $b$.

When we rationalize the denominator of $x = \frac { \sqrt { 5 } + 3 } { \sqrt { 5 } + 2 }$, we have

$$x = \sqrt { \mathbf { A } } - \mathbf { B }$$

Since this is a solution of equation (1), by substituting this in (1) we have

$$- a + b + \mathbf { C } + ( a - \mathbf { D } ) \sqrt { \mathbf { E } } = 0 .$$

Hence we see that

$$a = \mathbf { F } \text { and } b = \mathbf { G H } .$$

(2) Next, we are to find the smallest integer $p$ such that both solutions of equation (1) satisfy inequality (2).

When we solve inequality (2), we have

$$x > - p - 1 .$$

Since both solutions of equation (1) satisfy this, we see that

$$p > \sqrt { \mathbf { J } } - \mathbf { K } .$$

Hence the smallest integer $p$ is $\mathbf { L }$.