 Hello and welcome to the session. I am Asha and I am going to help you solve the following question that says, expand using the binomial theory 1 plus x upon 2 minus 2 upon x raised to the power 4, where x is not equal to 0. So let us now begin with the solution. Here we have to expand 1 plus x upon 2 minus 2 upon x raised to the power 4 or it can further be written as 1 plus x upon 2 minus 2 upon x raised to the power 4. Now expanding this with the binomial theorem it can further be written as 4 c 0 1 plus x upon 2 raised to the power 4 into minus 2 upon x raised to the power 0 plus 4 c 1 1 plus x upon 2 raised to the power 3 into minus 2 upon x raised to the power 1 4 c 2 1 plus x upon 2 raised to the power 2 into minus 2 upon x square plus 4 c 3 1 plus x upon 2 raised to the power 1 into minus 2 upon x raised to the power 3 plus 4 c 4 into 1 plus x upon 2 raised to the power 0 into minus 2 upon x raised to the power 4. This is further equal to 4 c 0 is 1. So 1 plus x upon 2 raised to the power 4 and minus 2 upon x raised to the power 0 is 1 plus 4 since 4 c 1 is 4 and let us expand 1 plus x upon 2 whole cube. So this is 1 cube plus 3 into 1 square into x upon 2 plus 3 into 1 into x upon 2 whole square plus x upon 2 whole cube into minus 2 upon x plus 4 c 2 6 1 plus x upon 2 whole square is 1 plus 2 into 1 into x upon 2 plus x upon 2 whole square into minus 2 upon x whole square is 4 upon x square plus 4 c 4 into 1 plus x upon 2 minus 8 upon x cube plus 4 c 4 is 1 and 1 plus x upon 2 raised to the power 0 is 1 and here we have 16 upon x raised to the power 4. Now we will expand 1 plus x upon 2 raised to the power 4 with the help of binomial theorem. So this will be equal to 4 c 0 1 raised to the power 4 into x upon 2 raised to the power 0 plus 4 c 1 1 raised to the power 3 upon x into x upon 2 raised to the power 1 plus 4 c 2 1 raised to the power 2 x upon 2 raised to the power 2 plus 4 c 3 1 raised to the power 1 into x upon 2 raised to the power 3 plus 4 c 4 1 raised to the power 0 into x upon 2 raised to the power 4 plus now opening this bracket 4 into 1 is 4 plus 2 cancels out with 4 so we have 2 into 3 6 into x plus here we have 3 upon 4 so 4 cancels out with 4 and we have 3 x square and on simplifying we have x cube upon 2 and multiplying these 4 terms with minus 2 upon x plus now simplifying this bracket we have 6 plus x plus x square upon 4 into 6 here also we have 6 on multiplying 6 by 2 term of this bracket into 4 upon x square plus 4 plus 2 into x into minus 8 upon x cube plus 16 upon x raised to the power 4 which is further equal to 4 c 0 is 1 and x upon 2 raised to the power 0 is also 1 and this is also 1 so we have 1 plus 4 c 1 is 4 into x upon 2 plus 4 c 2 is 6 to x square upon 4 4 c 3 is 4 into x cube upon 8 plus 4 c 4 is 1 we have x raised to the power 4 upon 16 now multiplying minus 2 upon x with this whole bracket we have multiplying with 4 we get minus 4 to the power 8 upon x now multiplying minus 2 upon x with 6 x we have plus into minus is minus 6 to the power 12 now multiplying 3 x square with this bracket we have minus 6 into x and lastly multiplying x cube upon 2 with this bracket we get minus x square now multiplying these two brackets we have 6 4 to the power 24 upon x square plus 6 4 to the power 24 upon x plus here 4 cancels out with 4 x square with x so we have only 6 plus lastly simplifying this bracket we have 8 into 4 is 32 with a negative sign upon x cube minus 8 to the power 16 upon x square plus 16 upon x raised to the power 4 which is further equal to 1 plus 2 x plus 3 upon 2 x square plus x cube upon 2 plus x raised to the power 4 upon 16 minus 8 upon x minus 12 minus 6 x minus x square plus 24 upon x square plus 24 upon x plus 6 minus 32 upon x cube minus 16 upon x square plus 16 upon x raised to the power 4 this is further equal to combining the light terms 1 minus 12 and plus x gives minus 5 plus 2 x and minus 6 x gives minus 4 x 3 upon 2 x square and minus x square gets plus x square upon 2 plus x cube upon 2 as it is x 4 upon 16 as it is and we have minus 8 upon x and plus 24 upon x so this is equal to plus 16 upon x then we have 24 upon x square minus 16 upon x square so this gets plus 8 upon x square and we have 24 upon x already joined with minus 8 upon x so we get 16 upon x now we are left with minus 32 upon x cube and lastly 16 upon x raised to the power 4 so it is handling the given term our answer is 16 upon x plus 8 upon x square minus 32 upon x cube plus 16 upon x raised to the power 4 minus 4 x plus x square upon 2 plus x cube upon 2 raised to the power 4 upon 16 minus 5 so this completes the session take care and have a good day.