 Hello and welcome to the session. In this session, we will discuss a question which says that, first part, find the product a cube minus a square plus 1 the goal into 2 a cube minus 5 the goal. Second part is, simplify b square plus b cube into b plus 1 the goal and third part is, find the product 2x minus 1 the goal into 3 minus x the whole. Now before starting the solution of this question, we should know a result and that is the product layer of exponents which is x raise to power n into x raise to power n is equal to x raise to power n plus n. Now this result will work out as a key idea for solving out the given question. Now let us start with the solution of the given question. Let us start with the first part and the first part we have to find the product a cube minus a square plus 1 the goal into 2 a cube minus 5 the goal. Now we will use distributive property to multiply the two polynomials. Here we will multiply each term of first polynomial by the second polynomial. So this is equal to a cube into 2 a cube minus 5 the goal minus a square into 2 a cube minus 5 the goal plus 1 into 2 a cube minus 5 the goal. Now to multiply the terms we will use the product law of exponents which we have discussed in the key idea. So this is equal to now a cube into 2 a cube will be 2 into a raise to power 3 plus 3 which is 6. So this will be 2 into a raise to power 6. Now a cube into minus 5 will be minus 5 a cube here minus a square into 2 a cube will be minus 2 into a raise to power 2 plus 3 which is 5. So this will be minus 2 into a raise to power 5. Then minus into minus is plus and a square into 5 is 5 a square plus 1 into 2 a cube is 2 a cube and 1 into minus 5 is minus 5. Now combining life terms we get now here minus 5 a cube and plus 2 a cube are life terms. So this will be 2 into a raise to power 6 minus 2 into a raise to power 5 plus of minus 5 a cube plus 2 a cube. The whole plus 5 a square minus 5. Now this is equal to 2 into a raise to power 6 minus 2 into a raise to power 5. Now minus 5 a cube plus 2 a cube is equal to minus 3 a cube and plus into minus is minus. So this will be minus 3 a cube plus 5 a square minus 5. So this is the required product and now let us start with second part. Now in the second part we have to simplify b square plus b cube into b plus 1 the whole. Here in bracket we have b plus 1 and only b cube is multiplied with it. So we open the brackets by multiplying the terms and using the product law of exponents. So this is equal to b square plus b cube into b plus b cube into 1 which is equal to b square plus. Now b cube into b will be b raise to power 3 plus 1 which will be b raise to power 4 plus b cube into 1 is b cube. Now here you can see but there are no like terms. So we cannot add these terms and this expression will remain as it is. And finally we will write this expression or you can say we will write this polynomial in descending order of powers of b. So this will be equal to b raise to power 4 plus b raise to power 3 plus b raise to power 2. So this is the required answer for the second part and now we will start with the third part. Now in the third part we have to find the product 2x minus 1 the whole into 3 minus x the whole. Now here we have to multiply the two binomials. So here we will use the same method which we have used in first part that is here we will multiply each term of first polynomial by the second polynomial. So this is equal to 2x into 3 minus x the whole minus 1 into 3 minus x the whole. However this is equal to now 2x into 3 is equal to 6x. 2x into minus x will be minus 2 into x raise to power 1 plus 1 versus 2. So it will be minus 2x square. Now minus 1 into 3 is minus 3 and minus 1 into minus x is plus x. And now we will combine the light terms. Now here 6x and x are light terms. So this will be equal to 6x plus x the whole minus 2x square minus 3. And this is equal to now 6x plus x is 7x minus 2x square minus 3. And now we will arrange this polynomial in the second order of powers of x. So this will be equal to minus 2x square plus 7x minus 3. And this is the required answer for the third part. So this completes our session. Hope you all have enjoyed this session.