To determine the total number of parallel tangents between the functions $f(x) = x^2 - x + 1$ and $g(x) = x^3 - x^2 - 2x + 1$, we can analyze the derivatives of these functions. Parallel tangents will occur where the slopes (derivatives) of the two functions are equal.

1. Find the derivative of $f(x)$: $f'(x) = \frac{d}{dx}(x^2 - x + 1) = 2x - 1$

2. Find the derivative of $g(x)$: $g'(x) = \frac{d}{dx}(x^3 - x^2 - 2x + 1) = 3x^2 - 2x - 2$

3. Set the derivatives equal to each other to find points where the slopes are the same: $2x - 1 = 3x^2 - 2x - 2$ Rearranging this equation gives: $3x^2 - 4x - 1 = 0$

4. Solve the quadratic equation $3x^2 - 4x - 1 = 0$ to find the values of $x$: Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 3$, $b = -4$, and $c = -1$: $x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 3 \cdot (-1)}}{2 \cdot 3}$ $x = \frac{4 \pm \sqrt{16 + 12}}{6}$ $x = \frac{4 \pm \sqrt{28}}{6}$ $x = \frac{4 \pm 2\sqrt{7}}{6}$ $x = \frac{2 \pm \sqrt{7}}{3}$ This provides two real values of $x$, meaning there are two points at which the tangents are parallel.

5. Conclusion: Since there are exactly two points where the slopes are equal, the answer is:

   (a) finite

Would you like a more detailed breakdown of any step or further clarification?

Here are five related questions for deeper understanding:

1. How do we find the equation of a tangent line at a given point on a curve?
2. Why does setting the derivatives equal determine parallel tangents?
3. How can we interpret the solutions of the quadratic equation in this context?
4. What is the geometric meaning of finding parallel tangents between two functions?
5. Can this approach be applied to functions of higher degrees, and if so, how?

Tip: When solving for parallel tangents, setting the derivatives equal allows you to find points with matching slopes, indicating tangents that are parallel.