Final answer:
To determine the missing side length of the isosceles triangle, subtract twice the length of one of the equal sides from the total perimeter, resulting in the expression 4x^4 - 4x^3 + 6x - 12.
Step-by-step explanation:
The student is given that an isosceles triangle has two equal sides with a length of 2x^3 + 7x^2 - 4x + 9, and the total perimeter of the triangle is P = 4x^4 + 14x^2 - 2x + 6. To find the length of the third side, we must subtract the combined lengths of the two equal sides from the total perimeter.
Let the length of the missing side be L, then:
L = P - 2 × (length of one equal side)
L = (4x^4 + 14x^2 - 2x + 6) - 2 × (2x^3 + 7x^2 - 4x + 9)
L = 4x^4 + 14x^2 - 2x + 6 - (4x^3 + 14x^2 - 8x + 18)
L = 4x^4 - 4x^3 + 14x^2 - 14x^2 - 2x + 8x + 6 - 18
L = 4x^4 - 4x^3 + 6x - 12 (This is the expression for the missing length).