Let's work through these questions step by step:
A. To find the total of two sides, simply add the lengths of the two given sides:
Total = (3x^2 + 2x - 1) + (4x - x^2 - 1)
Total = 3x^2 + 2x - 1 + 4x - x^2 - 1
Total = (3x^2 - x^2) + (2x + 4x) + (-1 - 1)
Total = 2x^2 + 6x - 2
B. If the perimeter of the triangle is given as x^3 - x^2 + 8x - 1, and you already know the total of two sides from part A (2x^2 + 6x - 2), you can find the length of the missing side by subtracting the total of two sides from the perimeter:
Missing Side = Perimeter - Total of Two Sides
Missing Side = (x^3 - x^2 + 8x - 1) - (2x^2 + 6x - 2)
Missing Side = x^3 - x^2 + 8x - 1 - 2x^2 - 6x + 2
Missing Side = (x^3 - 2x^2) + (-x^2 - 6x) + (8x - 1 + 2)
Missing Side = x^2(x - 2) - x(x + 6) + 9
C. If we know that x = 2, we can substitute this value into the expressions for the sides:
Side 1 = 3x^2 + 2x - 1 = 3(2^2) + 2(2) - 1 = 12 + 4 - 1 = 15
Side 2 = 4x - x^2 - 1 = 4(2) - 2^2 - 1 = 8 - 4 - 1 = 3
Missing Side = x^2(x - 2) - x(x + 6) + 9 = 2^2(2 - 2) - 2(2 + 6) + 9 = 0 - 16 + 9 = -7
So, when x = 2, the lengths of the three sides are: 15, 3, and -7.
D. Regarding the possibility of having these side lengths for a triangle, there is an issue. For a triangle to be valid, the sum of the lengths of any two sides must be greater than or equal to the length of the third side (Triangle Inequality Theorem).
In this case, if the missing side is -7, it's not possible to form a triangle because the sum of any two sides would be less than -7, which is not a valid length for a side. Therefore, these side lengths cannot form a triangle.