 Hi and welcome to the session. I am Asha and I am going to help you solve the following problem which says factorize and factorize these two problems. Let us first learn a simple identity which says if we have A square plus B square plus C square plus 2 into A into B plus 2 into B into C plus 2 into C into A then this is equal to A plus B plus C whole square. So, with this identity and key idea with the help of which we will solve these two problems. Let us now start with the solution and the first one is 4x square plus 9y square plus 16z square plus 12xy minus 24yz minus 16xz. Now, here 4x square can be written as 2x whole square, 9y square can be written as 3y whole square and 16z square can be written as 4z whole square. Here xy can be written as 2 into 2x into 3y, 24yz can be written as into 3y into 4z and 16xz can be written as 2 into 4z. The given problem which is 4x square plus 9y square plus 16z square plus 12xy minus 24yz minus 16xz can be written in the form of A square plus B square plus C square plus 2ab plus 2bc plus 2ca. Now, here A is equal to 2x, so we have 2x whole square, B is 3y whole square and C is 4z, so we have 4z whole square. Now, let us see these two terms. Now, both these terms are negative and the common literal in both these terms is z. So, z must be a negative term that is the literal z had the coefficient as negative, so here also we must have this point as negative. So, this is the main thing which you have to remember while observing these two problems, the number of signs if they are even and have the negative sign then we see the common literal and then we assign the negative sign to that term which have the common literal. So, further 12xy can be written as 2 into 2x into 3y then we have minus 24yz can be written as 2 into 3y into 4z and 16xz can be written as 2 into 2x into 4z. Now, further it can be written as 2x whole square plus 3y whole square plus minus 4z whole square plus 2 into 2x into 3y. Now, this is plus and since plus into minus is minus, so it can be written as 2 into 3y into minus 4z. Since minus 4z is a negative term and the last again writing in the same form plus 2 into 2x into minus 4z. Now, this is in the form of the left hand side of the identity where a is equal to 2x, b is equal to 3y and c is equal to minus 4z and that is when applying the identity this can further written as 2x plus 3y plus minus 4z whole square which is further equal to 2x plus 3y plus into minus is minus, so minus 4z whole square and this is 2x plus 3y minus 4z into 2x plus 3y minus 4z and hence the factors of the first one is 2x plus 3y minus 4z into 2x plus 3y minus 4z. So, this completes the first part and now proceeding on to the second part which is 2x square plus y square plus 8z square minus 2 root 2xy plus 4 root 2yz minus 8xz. Now, let us write down 2x square as root 2x whole square. Similarly, y square can be written as y whole square and 8z square can be written as 2 root 2z whole square. Now, observing both these negative terms we find that the common literal in these two terms is x, so we must have 2x square as minus root 2x whole square. Now, let us write down the rest of the terms. First we have 2 root 2xy which can be written as 2 into minus root 2x into y. This is also a negative term, so minus 2 root 2xy can be written as 2 into minus root 2x into y, 4 root 2yz can be written as 2 into y into 2 root 2z and the last term which is minus 8xz can be written as 2 times of minus root 2x into 2 root 2z. Now, the given problem which is 2x square plus y square plus 8z square minus 2 root 2xy plus 4 root 2yz minus 8xz can be written as root 2x with a negative sign whole square plus y whole square. 8z square can be written as 2 root 2z whole square plus minus 2 root 2xy can be written as 2 into minus root 2x into y plus 4 root 2yz can be written as 2 into y into 2 root 2z and the last term which is minus 8xz can be written as plus 2 into minus root 2x into 2 root 2z. Now, observing the identity we find here that a is equal to minus root 2x, b is equal to y and c is equal to 2 root 2z and that is the key idea plus it can be written as minus root 2x plus y plus 2 root 2z whole square this is by a key idea which is further equal to minus root 2x plus y plus 2 root 2z into minus root 2x plus y plus 2 root 2z and hence this is our answer so hope you enjoyed this session take care and have a good day.