 Hello and how are you all today? The question says if x minus 2x plus 3 is the hcf of the polynomials, px is equal to x square minus 3x plus 2 into 2x square plus 7x plus a and qx that is x square plus 4x plus 3 into 3x square minus 7x plus b, find the values of a and b. So let's proceed with the solution. Now here we have the given functions px is equal to x square minus 3x plus 2 into 2x square plus 7x plus a and we have qx equal to x square plus 4x plus 3 into 3x square minus 7x plus b, right? Further we have the hcf of these two polynomials as x minus 2 into x plus 3. So that means x minus 2 and x plus 3 are the factors of px and qx, right? So we have, so the value of this function when x is equal to 2 comes out to be 0 and when we put the value of x as minus 3 it is coming out to be 0 again. Also we have q, the value of the function q and the value of x is 2 coming out to be 0 and q when it is minus 3 is also 0. That is we have obtained all these things by using factor theory. Now let us find out the value of the function px when the value of x is 2 it is coming out to be 4 minus 6 plus 2 into 8 plus 14 plus a. Now we equate it to 0 that we have obtained it from factor theorem. This further implies that the value of a is coming out to be 3 and when we have the value of x as minus 2, sorry minus 3 and we substitute this value in q we have it as 9 minus 12 plus 3 into 27 plus 21 plus b and it should be equal to 0. This implies that the value of b is coming out to be 2, right? So the value of a equal to 3 and b equal to 2 is the required solution. So hope you understood it well and enjoyed it to have a nice day.