To solve this problem, we first need to understand the region bounded by the curves \(y = x^2\) and \(y = 2x^2 - 1\).

Step 1: Find Points of Intersection

First, we find the points of intersection between the curves:

\(x^2 = 2x^2 - 1\)

Rearranging gives:

\(x^2 - 1 = 0\)

Solving this equation, we find:

\(x = \pm 1\)

So, the points of intersection are \((-1, 1)\) and \((1, 1)\).

Step 2: Analyze the Partition by Two Lines

To determine how the lines partition the region \(R\), consider two distinct lines \(L_1\) and \(L_2\). The total number of regions \(n'\) created by two intersecting lines depends on their relative positions and intersections:

1. If two lines intersect within the region, they can divide it into 4 regions.
2. If these lines do not both intersect within the given boundary, they can still considerably divide the curve more.

For maximum partitioning, lines should intersect each other and additionally intersect the curves.

Step 3: Calculate Maximum Possible Regions

With two lines intersecting each other and both intersecting the boundary curves, we can partition the region into 6 distinct areas.

Conclusion

Considering the above possibilities, it's possible for the curves and the lines to create 6 distinct regions, satisfying the condition \(n'\) can be 6.

Correct Answer: 'n' can be 6.