We expand $(10 - x)^3 + (x - 2)^3$ as a sum of cubes, which states that \[(a + b)(a^2 - ab + b^2) = a^3 + b^3.\]Thus, let $a = 10 - x$ and $b = x - 2$. Then $(10 - x)^3 + (x - 2)^3 = (10 - x + x - 2)((10 - x)^2 - (10 - x)(x - 2) + (x - 2)^2)$.
Now, since $(10 - x)(x - 2) = 11$, we know that \[(10 - x + x - 2)((10 - x)^2 - (10 - x)(x - 2) + (x - 2)^2) = 8((10 - x)^2 - 11 + (x - 2)^2).\]But, notice that $(10 - x)^2 - 11 + (x - 2)^2$ is equal to \[(x^2 - 20x + 100) - 11 + (x^2 - 4x + 4) = 2x^2 - 24x + 93.\]Therefore, $8((10 - x)^2 - 11 + (x - 2)^2) = 8 \cdot (2x^2 - 24x + 93) = \boxed{16x^2 - 192x + 744}$.
If $(10 - x)(x - 2) = 11$, then compute $(10 - x)^3 + (x - 2)^3$.
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