 Hi and welcome to the session, let us discuss the following question today. On dividing x cube minus 3x square plus x plus 2 by a polynomial gx, the quotient and remainder were x minus 2 and minus 2x plus 4 respectively, phi gx. Let us now start the solution. We are given dividend is equal to x cube minus 3x square plus x plus 2, divisor is equal to gx, quotient is equal to x minus 2, remainder is equal to minus 2x plus 4. Now we know, dividend is equal to divisor multiplied by quotient plus remainder. So, substituting their corresponding values we get x cube minus 3x square plus x plus 2 is equal to gx multiplied by x minus 2 plus minus 2x plus 4. Now this implies x cube minus 3x square plus x plus 2 minus 2x plus 4 is equal to gx multiplied by x minus 2. Now this implies x cube minus 3x square plus 3x minus 2 is equal to gx multiplied by x minus 2. Now this implies gx is equal to x cube minus 3x square plus 3x minus 2 upon x minus 2. Now we will divide x cube minus 3x square plus 3x minus 2 by x minus 2 to get gx. Now to get x cube we will multiply the divisor by x square. So, we get x cube minus 2x square. Now we will subtract the like terms and rewrite the remaining terms of the dividend. We get minus x square plus 3x minus 2. Now we will multiply the divisor by minus x to get minus x square. So, we will write minus x minus x square plus 2x. Now subtracting the like terms and rewriting the rest of the terms of the dividend we get x minus 2 and x square and x square will get cancelled. Now we will multiply the divisor by 1 to get x minus 2. So, we will write plus 1. Now subtracting the right terms we get remainder as 0. Clearly we can see gx is equal to x square minus x plus 1. So, we can write gx is equal to x square minus x plus 1. So, this is our required answer. This completes the session. Hope you understood the session. Goodbye.