[tex] \frac{1}{(x - 1)(x - 2)} + \frac{3}{x - 1} = \frac{3 - x}{x - 2} [/tex]

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To solve the given equation, we need to find a common denominator for all the fractions on the left side.

The common denominator for (\frac{1}{(x-1)(x-2)}) and (\frac{3}{x-1}) is ((x-1)(x-2)).

Therefore, multiplying each fraction by the appropriate factor to get the common denominator, we get:

[\frac{1}{(x - 1)(x - 2)} \cdot \frac{x - 2}{x - 2} + \frac{3}{x - 1} \cdot \frac{(x - 2)}{(x-2)} = \frac{3 - x}{x - 2}]

[ \frac{x - 2}{(x - 1)(x - 2)} + \frac{3x - 6}{(x - 1)(x-2)} = \frac{3 - x}{x - 2}]

Now combine the fractions on the left side:

[\frac{x - 2 + 3x - 6}{(x - 1)(x - 2)} = \frac{3 - x}{x - 2}]

[\frac{4x - 8}{(x - 1)(x - 2)} = \frac{3 - x}{x - 2}]

Now cross multiply:

[(4x - 8)(x - 2) = (3 - x)(x - 1)]

Expand both sides:

[4x^2 - 8x - 8x + 16 = 3x - x^2 - 3 + x]

[4x^2 - 16x + 16 = 3x - x^2 - 3 + x]

Rearrange the equation:

[5x^2 - 19x + 19 = 0]

This is a quadratic equation. To solve for x, we can use the quadratic formula:

[x = \frac{-(-19) \pm \sqrt{(-19)^2 - 4(5)(19)}}{2(5)}]

[x = \frac{19 \pm \sqrt{361 - 380}}{10}]

[x = \frac{19 \pm \sqrt{-19}}{10}]

[x = \frac{19 \pm i\sqrt{19}}{10}]

Therefore, the solutions to the given equation are:

[x = \frac{19 + i\sqrt{19}}{10}, \frac{19 - i\sqrt{19}}{10}]