Verified answer

If 4 - 5i is a zero then 4 + 5i must also be a zero.

Use the factor theorem to convert the four zeros into factors:

(x - 4 + 5i) (x - 4 - 5i) (x - 5) (x - 5)

Multiply these factors together:

x^4 - 18x^3 + 146x^2 - 610x + 1025

[Yours looks like a calculation error when doing the multiplication.]

Since 4 - 5i is a zero, so also is 4 + 5i since such zeros occur in conjugate pairs. Now the sum of these complex roots is 8 and their product is 4² - (5i)² = 41. Thus they are the roots of the quadratic x² -8x + 41 = 0, which must therefore be one factor of the required quartic. (x - 5)² being the other factor, the quartic is (x² - 8x +41)(x² -10x + 25) = x⁴-10x³ + 25x² - 8x³+ 80x² - 200x + 41x² - 410x + 1025 = x⁴-18x³ + 146x² - 610x + 1025.