The distance from Newtown to Oldtown on the highway is (6x^2+2x-2) miles. Using the back roads, the distance is (5x^2-8x-6) miles. How many miles shorter is the second route?
To find the difference between the two distances, we need to subtract the second distance (back roads) from the first distance (highway):
(6x^2+2x-2) - (5x^2-8x-6)
Simplifying this expression, we can combine like terms:
6x^2 - 5x^2 + 2x + 8x - 2 - (-6)
The -(-6) becomes +6:
x^2 + 10x + 4
This is the simplified expression for the difference between the two distances. We cannot simplify this any further. Therefore, the answer is:
x^2 + 10x + 4 miles shorter on the back roads.
To find how many miles shorter the second route is, subtract the distance of the second route from that of the first route:
(6x^2+2x-2) - (5x^2-8x-6)= 6x^2+2x-2 - 5x^2 + 8x + 6 = x^2 + 10x + 4 miles. Answer: \boxed{x^2 + 10x + 4}.