If the range of the function $f(x) = \frac{5 - x}{x^2 - 3x + 2}$, $x \neq 1{,}2$, is $( -\infty, \alpha) \cup [\beta, \infty)$, then $\alpha^2 + \beta^2$ is equal to:

Step 1: Rewrite the function and set up for range calculation

To find the range of the function, we first set $y = f(x)$. The denominator can be factored as $x^2 - 3x + 2 = (x-1)(x-2)$. So the function is $y = \frac{5-x}{(x-1)(x-2)}$. We need to express $x$ in terms of $y$ to find the possible values of $y$.

Step 2: Rearrange into a quadratic equation in x

Multiply both sides by the denominator $(x-1)(x-2)$ to get $y(x^2 - 3x + 2) = 5 - x$. Rearranging the terms to form a quadratic equation in $x$, we get $yx^2 - 3yx + 2y = 5 - x$, which simplifies to $yx^2 + (-3y+1)x + (2y-5) = 0$.

Step 3: Apply the discriminant condition for real roots

For $x$ to be a real number, the discriminant ($D$) of this quadratic equation must be greater than or equal to zero. The discriminant is given by $D = b^2 - 4ac$, where $a=y$, $b=(-3y+1)$, and $c=(2y-5)$.

Step 4: Solve the inequality for y

Expanding the discriminant, we get $(9y^2 - 6y + 1) - (8y^2 - 20y) \ge 0$. This simplifies to $9y^2 - 6y + 1 - 8y^2 + 20y \ge 0$, which further simplifies to $y^2 + 14y + 1 \ge 0$.

Step 5: Find the roots of the quadratic and determine the range

To solve $y^2 + 14y + 1 \ge 0$, we find the roots of $y^2 + 14y + 1 = 0$ using the quadratic formula. The roots are $y = -7 + 4\sqrt{3}$ and $y = -7 - 4\sqrt{3}$. Since the parabola $y^2 + 14y + 1$ opens upwards, the inequality $y^2 + 14y + 1 \ge 0$ holds when $y \le -7 - 4\sqrt{3}$ or $y \ge -7 + 4\sqrt{3}$. Thus, $\alpha = -7 - 4\sqrt{3}$ and $\beta = -7 + 4\sqrt{3}$.

Step 6: Calculate $\alpha^2 + \beta^2$

Now we need to calculate $\alpha^2 + \beta^2$. We have $\alpha = -7 - 4\sqrt{3}$ and $\beta = -7 + 4\sqrt{3}$. Using the identity $(a-b)^2 + (a+b)^2 = 2(a^2+b^2)$, with $a=-7$ and $b=4\sqrt{3}$, we can simplify the calculation.

Step 7: Simplify the expression

Substituting the values into the identity, we get $2((-7)^2 + (4\sqrt{3})^2) = 2(49 + 16 \times 3) = 2(49 + 48) = 2(97) = 194$.

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