1. **State the problem:** Simplify the given complex algebraic expression involving variables $t_1$, $t_2$, $r_1$, $r_2$, $\epsilon_1$, $\epsilon_2$, and $x$.
2. **Identify the expression:** The expression is
$$
\frac{(t_1 - t_2)^2 \big(-2 r_1 t_1 t_2 \epsilon_1 \epsilon_2 - 2 r_2 t_1 t_2 \epsilon_1 \epsilon_2 + 2 t_1 t_2 x \epsilon_1 \epsilon_2 + t_1 t_2 \epsilon_1^2 \epsilon_2 + t_1 t_2 \epsilon_1 \epsilon_2^2 + t_1 \epsilon_1 + t_2 \epsilon_2\big) \sqrt{\cdots} + (t_1^2 - 2 t_1 t_2 + t_2^2) \big(\cdots\big)}{2 \epsilon_1^2 \epsilon_2^2 (t_1 - t_2)^2 (t_1^2 - 2 t_1 t_2 + t_2^2)}
$$
where $\sqrt{\cdots}$ and $(\cdots)$ represent large polynomial expressions.
3. **Note:** The denominator contains factors $(t_1 - t_2)^2$ and $(t_1^2 - 2 t_1 t_2 + t_2^2)$, but observe that
$$
t_1^2 - 2 t_1 t_2 + t_2^2 = (t_1 - t_2)^2
$$
so the denominator simplifies to
$$
2 \epsilon_1^2 \epsilon_2^2 (t_1 - t_2)^2 (t_1 - t_2)^2 = 2 \epsilon_1^2 \epsilon_2^2 (t_1 - t_2)^4
$$
4. **Rewrite denominator:**
$$
\text{Denominator} = 2 \epsilon_1^2 \epsilon_2^2 (t_1 - t_2)^4
$$
5. **Rewrite numerator:**
$$
\text{Numerator} = (t_1 - t_2)^2 A \sqrt{B} + (t_1 - t_2)^2 C
$$
where
$$
A = -2 r_1 t_1 t_2 \epsilon_1 \epsilon_2 - 2 r_2 t_1 t_2 \epsilon_1 \epsilon_2 + 2 t_1 t_2 x \epsilon_1 \epsilon_2 + t_1 t_2 \epsilon_1^2 \epsilon_2 + t_1 t_2 \epsilon_1 \epsilon_2^2 + t_1 \epsilon_1 + t_2 \epsilon_2
$$
and
$$
C = \text{the large polynomial inside the second big parentheses}
$$
6. **Factor out $(t_1 - t_2)^2$ from numerator:**
$$
\text{Numerator} = (t_1 - t_2)^2 (A \sqrt{B} + C)
$$
7. **Cancel $(t_1 - t_2)^2$ from numerator and denominator:**
$$
\frac{\cancel{(t_1 - t_2)^2} (A \sqrt{B} + C)}{2 \epsilon_1^2 \epsilon_2^2 \cancel{(t_1 - t_2)^4}} = \frac{A \sqrt{B} + C}{2 \epsilon_1^2 \epsilon_2^2 (t_1 - t_2)^2}
$$
8. **Final simplified form:**
$$
\boxed{\frac{A \sqrt{B} + C}{2 \epsilon_1^2 \epsilon_2^2 (t_1 - t_2)^2}}
$$
where $A$, $B$, and $C$ are as defined above.
**Summary:** The original expression simplifies by recognizing the perfect square in the denominator and canceling common factors, resulting in a cleaner fraction with denominator $2 \epsilon_1^2 \epsilon_2^2 (t_1 - t_2)^2$ and numerator $A \sqrt{B} + C$.
This is the most straightforward simplification without expanding the large polynomials further, which would be impractical here.
Expression Simplification Ac14E7
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