Let m be an odd integer. Show that the number racm4 + 4m2 + 1116 is an integer.

Since $m$ is an odd integer, we can write $m = 2k + 1$ for some integer $k$.

Since $k(k+1)$ is always an even integer (the product of two consecutive integers), let $k(k+1) = 2j$ for some integer $j$.
Then, $m^2 = 4(2j) + 1 = 8j + 1$.

The numerator is $m^4 + 4m^2 + 11$.

Since $j$ is an integer, $4j^2 + 3j + 1$ is also an integer.
Therefore, the number $\frac{m^4 + 4m^2 + 11}{16}$ is an integer.

2. If $a, b,$ and $c$ are positive real numbers, then show that: $\sqrt{\frac{b}{a}} \cdot \sqrt{\frac{c}{b}} \cdot \sqrt{\frac{a}{c}} = 1$

Since $a, b, c$ are positive real numbers, we can use the property $\sqrt{x} \cdot \sqrt{y} \cdot \sqrt{z} = \sqrt{xyz}$.

Cancel out the common terms $a, b, c$ from the numerator and denominator.

Thus, $\boxed{\sqrt{\frac{b}{a}} \cdot \sqrt{\frac{c}{b}} \cdot \sqrt{\frac{a}{c}} = 1}$.

3. Suppose $\sqrt{n} = \frac{3}{2}$, find $-\sqrt{n}$

We are given that $\sqrt{n} = \frac{3}{2}$.
We need to find the value of $-\sqrt{n}$.

The value of $-\sqrt{n}$ is $\boxed{-\frac{3}{2}}$.

4. Let $a$ and $b$ be positive real numbers. Show that $\frac{a}{b} + \frac{b}{a} \ge 2$

For any two positive real numbers $x$ and $y$, the Arithmetic Mean-Geometric Mean (AM-GM) inequality states that $\