If the tangent to the curve $$y = x^3 - x^2 + x$$ at the point (a, b) is also tangent to the curve $$y = 5x^2 + 2x - 25$$ at the point (2, -1), then $$|2a + 9b|$$ is equal to _____

We have the curve $$y = x^3 - x^2 + x$$ and we need the tangent at $$(a, b)$$ on this curve that is also tangent to $$y = 5x^2 + 2x - 25$$ at the point $$(2, -1)$$.

First, we verify $$(2, -1)$$ lies on the second curve: $$5(4) + 2(2) - 25 = 20 + 4 - 25 = -1$$. Yes.

The slope of the tangent to $$y = 5x^2 + 2x - 25$$ at $$x = 2$$ is $$y' = 10x + 2 = 22$$. So the tangent line at $$(2, -1)$$ is $$y - (-1) = 22(x - 2)$$, giving $$y = 22x - 45$$.

This same line is tangent to $$y = x^3 - x^2 + x$$ at $$(a, b)$$. Since $$(a,b)$$ is on the first curve, $$b = a^3 - a^2 + a$$. The slope at $$x = a$$ is $$3a^2 - 2a + 1 = 22$$, so $$3a^2 - 2a - 21 = 0$$. Using the quadratic formula: $$a = \frac{2 \pm \sqrt{4 + 252}}{6} = \frac{2 \pm 16}{6}$$, giving $$a = 3$$ or $$a = -\frac{7}{3}$$.

We check which value gives a tangent line passing through $$(2, -1)$$. The tangent at $$(a, b)$$ is $$y - b = 22(x - a)$$.

Case 1: $$a = 3$$: $$b = 27 - 9 + 3 = 21$$. Tangent: $$y - 21 = 22(x - 3)$$, so $$y = 22x - 45$$. At $$x = 2$$: $$y = 44 - 45 = -1$$. This matches.

Case 2: $$a = -\frac{7}{3}$$: $$b = -\frac{343}{27} - \frac{49}{9} - \frac{7}{3} = -\frac{343}{27} - \frac{147}{27} - \frac{63}{27} = -\frac{553}{27}$$. Tangent: $$y + \frac{553}{27} = 22\!\left(x + \frac{7}{3}\right)$$. At $$x = 2$$: $$y = 22 \cdot 2 + \frac{154}{3} - \frac{553}{27} = 44 + \frac{1386}{27} - \frac{553}{27} = 44 + \frac{833}{27} \neq -1$$.

So $$a = 3, b = 21$$. Therefore $$|2a + 9b| = |6 + 189| = 195$$.

Hence, the correct answer is $$\boxed{195}$$.