 Hello and welcome to the session. The given question says the hc-philsyme of two polynomials px and qx are 2x-1 and 6x-q plus 25x-square minus 24x plus 5 respectively if px is equal to 2x-square plus 9x-5 determine qx. Let's start with the solution and we are given a polynomial px which is equal to 2x-square plus 9x-5 and we have to find the other polynomial qx such that lcm of x and qx is given to us as 6x-q plus 25x-square minus 24x plus 5 and hcf px and qx is equal to 2x-1 and we have to find the polynomial qx. Now we know that the product of polynomial px and qx is equal to the product of the lcm and hcf so this implies qx is equal to lcm into hcf divided by px. Let's now put the values of lcm hcf as 6x-q plus 25x-square minus 24x plus 5 hcf is 2x-1 whole divided by px which is 2x-square plus 9x-5. Now let us factorize the denominator first. The denominator is 9x-5 this can be written as 2x-square plus 10x minus x-5 by splitting the middle term. Now taking 2x common from the first two terms and minus 1 common from the last two terms we have 2x into x plus 5 minus 1 into x plus 5 and this is equal to 2x minus 1 into x plus 5. So here the denominator on factorizing we get 2x-1 into x plus 5 and in the numerator we have 6x-q plus 25x-square minus 24x plus 5 into 2x-1 and now 2x-1 is the common factor. So we have 6x-q plus 25x-square minus 24x plus 5 whole divided by x plus 5. Now x plus 5 is a factor of the numerator therefore it can be written as x plus 5 into 6x-square minus 5x plus 1 whole divided by x plus 5. Now on canceling x plus 5 with x plus 5 we have 6x-square minus 5x plus 1 and this is the polynomial qx. Hence our answer is qx is 6x-square minus 5x plus 1. So this completes the session by intake care.