On dividing 3x^{3} + x^{2} + 2x + 5 by a polynomial g(X), the quotient and remainder are (3x – 5) and (9x + 10) respectively. Find g(x).

We know that,

According to the division rule, we have

Dividend = Quotient × Divisor + Remainder

Here, we have:

Dividend = 3x^{3} + x^{2} + 2x + 5

Quotient = 3x – 5

Remainder = 9x + 10

Putting these values in the formula, we get

3x^{3} + x^{2} + 2x + 5 = 3x – 5 × g (x) + 9x + 10

3x^{3} + x^{2} + 2x + 5 – 9x – 10 = (3x – 5) × g(x)

3x^{3} + x^{2} – 7x – 5 = (3x – 5) × g (x)

g (x) =

∴ g (x) = x^{2} + 2x + 1

10