 Hi and welcome to the session. I am Asha and I am going to help you solve the following problem which says fact wise, let us start with the solution. The first one is 12 x square minus 7 x plus 1. Now we will factorize this polynomial with the help of splitting the middle term. Now we have to split 12 such that the sum comes out to be 7. Now 12 is 2 6 is 12, 2 3 is 6 that is the factors of 12 are 2, 2 and 3 and this can be written as 4 into 3 and the sum of 4 and 3 is equal to 7 that this can further be written as 12 x square minus 7 can be written as 4 plus 3 into x plus 1. So this we have got by splitting the middle term. Now in opening the brackets we have 12 x square minus 4 x minus 3 x plus 1 and now taking 4 x common from the first two terms and minus 1 common from the last two terms we have 4 x and in the bracket we have 3 x minus 1. Now taking minus 1 common from the next two terms we have 3 x minus 1. Now taking 3 x minus 1 common we have 3 x minus 1 into 4 x minus 1 and hence on factorizing the given polynomial get the factor is 3 x minus 1 to 4 x minus 1. So this completes the first part let us now proceed on to the second part 2 x square plus 7 x plus 3. Now this can be written as 2 x square plus 7 can be written as 6 plus 1 into x plus 3 then 7 is equal to 6 plus 1 and 6 into 1 is equal to 2 into 3 where 2 is the coefficient of x square and 3 is the constant. Now this is further equal to 2 x square plus 6 x plus x plus 3 and now taking 2 x common from the first two terms and 1 common from the last two terms we have 2 x into x plus 3. Now taking 1 common from the last two terms we have x plus 3 in the next bracket. Now taking x plus 3 common we have x plus 3 into 2 x plus 1 plus on factorizing the second part we get the factors as x plus 3 into 2 x plus 1. So this completes the second part and now proceeding on to the third part where we have to factorize 6 x square plus 5 x minus 6 plus 6 x plus 3. Here again we have to split the middle term 5 in such a way that the product of these two numbers comes equal to 6 into 6 that is 36. Now on splitting 36 we have the factors as 2 into 18, 2 into 9 and 3 into 3. So 36 can be written as 2 into 2 into 3 into 3 or 4 into 9. When we subtract 4 from 9 we get the result as 5. Thus this statement can further be written as 6 x square plus 9 minus 4 into x minus 6 which is further equal to 6 x square plus 9 x minus 4 x minus 6 and now taking 3 x common from the first two terms and minus 2 common from the next two terms this can further be written as 3 x and what are we left with the first term that is 2 x plus and taking 3 x outside what are we left with 3 and taking minus 2 common we have 2 x plus 3 the last bracket. Now taking 2 x plus 3 common we have 2 x plus 3 into 3 x minus 2 and hence on factorizing the third polynomial we get the factors as 2 x plus 3 into 3 x minus 2. So this completes the third part and now proceeding on to the last one which is 3 x square minus x minus 4. Now this is quite simple 3 x square minus 1 x minus 4 and we have to split the middle term and 1 can be written as 4 minus 3 into x minus 4 and on opening the brackets we have 3 x square minus 4 x minus into minus is plus so plus 3 x minus 4. Now taking x common from the first two terms and 1 common from the last two terms it can further be written as x into 3 x minus 4 plus 1 into 3 x minus 4. Now taking 3 x minus 4 common get 3 x minus 4 and here we are left with x and here plus 1 and that is on factorizing the last polynomial we get the factors as x plus 1 3 x minus 4. So this completes the fourth part and hence the solution hope you enjoyed this session take care and have a good day.