 Hello and welcome to the session the given question says if x-2 into x plus 3 is the GCD of polynomials Px and Qx find the values of A and B. Let's start with the solution. Here we are given that GCD of Px and Qx is equal to x-2 into x plus 3. Now Px is equal to x-2 minus 3x plus 2 into Ax-2 plus 7x plus 3. Now the first term which is x-2 minus 3x plus 2 can be written as x-2 minus x minus 2x plus 2 by splitting the middle 2. Now taking x common from the first 2 terms and minus 2 common from the last 2 terms we have x into x minus 1 minus 2 into x minus 1. So this is equal to x minus 1 into x minus 2. So Px can further be written as x minus 1 into x minus 2 into Ax-2 plus 7x plus 3. Now we are given that x minus 2 into x plus 3 is the GCD of polynomials Px and Qx. So this implies when x is equal to 2 and minus 3 the value of this polynomial is 0. Therefore we have P at minus 3 is equal to 0. Now on replacing x by minus 3 in this equation we have minus 4 into minus 5 into 9A minus 21 plus 3 is equal to 0. Or we have 20 into 9A minus 18 is equal to 0 or this further implies that 9A is equal to 18 or A is equal to 2. Now we are given that the polynomial Qx is equal to 3x square plus 8x minus 3 into x square plus Bx plus 6. Now the polynomial 3x square plus 8x minus 3 can be written as 3x square plus 9x minus x minus 3. Now taking 3x common from the first 2 terms and minus 1 common from the last 2 terms we have 3x into x plus 3 minus 1 into x plus 3 and this is equal to 3x minus 1 into x plus 3. Therefore this can be written as 3x minus 1 into x plus 3 into x square plus Bx plus 6. Now as we are given that x minus 2 into x plus 3 is the GCD of the polynomials Px and Qx. Therefore Q at the point 2 is equal to 0. Now replacing x by 2 we have 5 into 5 into 4 plus 2 into B plus 6 is equal to 0 and this implies that 2B plus 10 is equal to 0 or B is equal to 10. Taking on the right hand side we have minus 10 divided by 2 and this gives us minus 5. Therefore B is equal to minus 5. Hence our answer is the value of A is 2 and the value of B is minus 5. So this completes the session. Hope you have understood it well. Bye and take care.