 Hello and welcome to the session. In this session we discuss the following question which says, factorize the following 6x to the power 6 minus 21x4 minus 12x square and minus 12a5 minus a cube plus a. Let's move on to the solution. The first expression given to us is 6x6 minus 21x4 minus 12x square. From this expression we take 3x square common and inside we have 2x4 minus 7x square minus 4. Consider the expression 2x4 minus 7x square minus 4. We need to find two numbers whose product is 2 into minus 4 that is minus 8 and some is minus 7. So now the two numbers are minus 8 and 1 since minus 8 into 1 is minus 8 and minus 8 plus 1 is minus 7. So we write this expression 2x4 minus 7x square minus 4 as 2x4 minus 8x square plus x square minus 4. Now this we have split at the middle term of this expression. So now here we have 3x square into 2x4 minus 8x square plus x square minus 4. Now we shall factorize this. So this is equal to 3x square. Now from this group we take out 2x square common inside we have x square minus 4. From this group we take out 1 common inside we have x square minus 4. Now this becomes further equal to 3x square into x square minus 4 into 2x square plus 1. We know that a square minus b square is a plus b into a minus b. So we can write x square minus 4 that is x square minus 2 square as x plus 2 into x minus 2. Hence we get this is equal to 3x square into x plus 2 into x minus 2 into 2x square plus 1. So the given expression 6x to the power 6 minus 21 x4 minus 12x square is equal to 3x square into x plus 2 into x minus 2 into 2x square plus 1. Now the second expression given to us is minus 12 a5 minus a cube plus a. In this we take out minus a common from this expression. So we have 12 a4 plus a square minus 1. Now for the expression 12 a4 plus a square minus 1 we will split the middle term that is plus a square. Now for this we need to find two numbers whose product is 12 into minus 1 that is minus 12 and sum is the coefficient of a square that is 1. Now here the two numbers are 4 and minus 3. 4 into minus 3 is minus 12 and 4 plus minus 3 is 1. So we write the given expression 12 a4 plus a square minus 1 as 12 a4 plus 4 a square minus 3 a square minus 1. So this is equal to minus a into 12 a4 plus 4 a square minus 3 a square minus 1. So this is one group, this is one group. Now this is equal to minus a. Now we take out 4 a square common from this first group. So we have 3 a square plus 1. Then from this group we take out minus 1 common inside we have 3 a square plus 1. So this expression is equal to minus a into 3 a square plus 1 into 4 a square minus 1. We know that a square minus b square is equal to a plus b into a minus b. So the expression 4 a square minus 1 which can be written as 2 a whole square minus 1 square is equal to 2 a plus 1 into 2 a minus 1. So this expression is equal to minus a into 3 a square plus 1 into 2 a plus 1 into 2 a minus 1. So the expression minus 12 a5 minus aq plus a is equal to minus a into 3 a square plus 1 into 2 a plus 1 into 2 a minus 1. Hence our final answer is for the first expression we have 3 x square into 2 x square plus 1 into x plus 2 into x minus 2. For the second expression we have minus a into 3 a square plus 1 into 2 a plus 1 into 2 a minus 1. This completes this session. Hope you have understood the solutions for this question.