1. The problem states that the roots of the cubic equation $$x^3 - 2x^2 + 4x - 5 = 0$$ are $$p$$, $$q$$, and $$r$$. We need to find the value of $$(p - 4)(q - 4)(r - 4)$$ without solving for the roots explicitly. 2. Recall the factorization of a cubic polynomial in terms of its roots: $$x^3 - 2x^2 + 4x - 5 = (x - p)(x - q)(x - r)$$ 3. To find $$(p - 4)(q - 4)(r - 4)$$, notice that this expression can be rewritten as: $$(p - 4)(q - 4)(r - 4) = - (4 - p)(4 - q)(4 - r)$$ But it is easier to consider the polynomial evaluated at $$x = 4$$: 4. Substitute $$x = 4$$ into the polynomial: $$f(4) = 4^3 - 2 \times 4^2 + 4 \times 4 - 5 = 64 - 32 + 16 - 5 = 43$$ 5. Since $$f(x) = (x - p)(x - q)(x - r)$$, then: $$f(4) = (4 - p)(4 - q)(4 - r) = 43$$ 6. Therefore: $$(p - 4)(q - 4)(r - 4) = - (4 - p)(4 - q)(4 - r) = -43$$ Final answer: $$(p - 4)(q - 4)(r - 4) = -43$$