On April 1st, in a class during math hour, they were practicing algebraic transformations. According to the task, they had to simplify the following fraction:

$$
\frac{(x+2)^{3}+(y+x)^{3}}{(x+2)^{3}-(y-2)^{3}}
$$

Ági, who is the best in math in the class, was being playful, so she said that if the denominator is not zero, then we can "simplify" by the exponents of three, meaning the result would be:

$$
\frac{x+2+y+x}{x+2-(y-2)}=\frac{2 x+y+2}{x-y+4}
$$

Decide whether the final result is correct.
Solution. Apply the following identities:

$$
a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right), \quad \text { and } \quad a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)
$$

then perform the operations in the fraction to be simplified:

$$
\begin{aligned}
& \frac{(x+2)^{3}+(y+x)^{3}}{(x+2)^{3}-(y-2)^{3}}=\frac{(x+2+y+x) \cdot\left[(x+2)^{2}-(x+2)(y+x)+(y+x)^{2}\right]}{(x+2-y+2) \cdot\left[(x+2)^{2}+(x+2)(y-2)+(y-2)^{2}\right]}= \\
& =\frac{(2 x+y+2)\left(x^{2}+4 x+4-x y-x^{2}-2 y-2 x+y^{2}+2 x y+x^{2}\right)}{(x-y+4)\left(x^{2}+4 x+4+x y+2 y-2 x-4+y^{2}-4 y+4\right)}= \\
& =\frac{2 x+y+2}{x-y+4} \cdot \frac{x^{2}+2 x+4+x y-2 y+y^{2}}{x^{2}+2 x+4+x y-2 y+y^{2}}=\frac{2 x+y+2}{x-y+4}
\end{aligned}
$$

Ági "had a joke in mind," but the final result is correct. Of course, in general, such simplification is not possible.

Remarks. 1. Lilla Szilágyi also noticed that Ági could have "joked around" even if a 2 was replaced by an arbitrary $b$ in the problem:

$$
\begin{aligned}
& \frac{(x+b)^{3}+(y+x)^{3}}{(x+b)^{3}-(y-b)^{3}} \\
= & \frac{(x+b+y+x) \cdot\left(x^{2}+2 b x+b^{2}-x y-b y-x^{2}-b x+y^{2}+2 x y+x^{2}\right)}{(x+b-y+x) \cdot\left(x^{2}+2 b x+b^{2}+x y+b y-b x-b^{2}+y^{2}-2 x y+b^{2}\right)}= \\
= & \frac{2 x+y+b}{x-y+2 b} \cdot \frac{x^{2}+b x+b^{2}+x y-b y+y^{2}}{x^{2}+b x+b^{2}+x y-b y+y^{2}}=\frac{2 x+y+b}{x-y+2 b} .
\end{aligned}
$$

The possibility of such a transformation is based on the fact that

$$
\frac{A^{3}+B^{3}}{A^{3}-(B-A)^{3}}=\frac{A+B}{A-(B-A)}=\frac{A+B}{2 A-B}
$$

2. The problem statement sparked the solvers' imagination. Some noticed that April 1st fell on a Sunday, so Ági and her classmates could not have practiced algebraic transformations in math class. Of course, these remarks are not valid, as the problem statement did not specify the year in which the class took place.