x = 2 + 3i

x – 2 = 3i

(x – 2)^{2} = 9i^{2}

x^{2} – 4x + 4 = 9(-1) …..**[∵ i**^{2} = -1]

x^{2} – 4x + 13 = 0 ….**.(i)**

Dividend = Divisor × Quotient + Remainder

∴ x^{3} – x^{2} + x + 46 = (x^{2} – 4x + 13) (x + 3) + 7

= 0(x + 3) + 7 ….**.[from(i)]**

= 7

**Alternate Method:**

x = 2 + 3i

α = 2 + 3i, \(\bar \alpha\) = 2 – 3i

α \(\bar \alpha\)= (2 + 3i)(2 – 3i)

= 4 – 6i + 6i – 9i^{2}

= 4 - 9(-1)

= 4 + 9

= 13

α + \(\bar \alpha\) = 2 + 3i + 2 - 3i = 4

∴ Standard form of quadratic equation,

x^{2} – (Sum of roots) x + Product of roots = 0

x^{2} – 4x + 13 = 0

Dividend = Divisor × Quotient + Remainder

∴ x^{3} – x^{2} + x + 46 = (x^{2} – 4x + 13).(x + 3) + 7

= 0(x + 3) + 7 …..**[From (i)]**

= 7