 Hello and welcome to the session. In this session, we will discuss the question which says that, to find the indicated operation, first part is x minus 3 whole upon 2x square minus 8 plus x plus 1 whole upon x plus 2 and second part is 3 upon x plus x upon 3x plus 4. Now let us start with the solution of the given question. Let us start with the first part. Now here we have to add the two rational expressions. Now here let us see factors in the denominators. Now in the first expression the denominator is 2x square minus 8. Now let us factorize it. Now this will be equal to taking two common from both the terms. It will be 2 into x square minus 4 double. Now we know that a square minus b square is equal to 8 plus b double into a minus b double. Now this is equal to 2 into x square minus 2 square double which is equal to now applying the formula of a square minus b square. This will be 2 into x plus 2 double into x minus 2 double. So on factorization denominator in first expression will be equal to 2 into x plus 2 the whole into x minus 2 the whole. Now where we can see that denominator in second expression is x plus 2. And now let us find lc of the denominators. Since the lc of denominators is product of all the factors in one of both the denominators. And here the repeated factors are taken only once in lcn. So lc of denominators here will be equal to 2 into. Now here you can see that x plus 2 is the repeated factor. So it will be taken only once. So it will be 2 into x plus 2 the whole into x minus 2 the whole. And now we will make the denominators equivalent in both the expressions using lcm. Now the first expression has denominator that is 2x square minus 8 equal to lcm that is 2 into x plus 2 the whole into x minus 2 the whole. So it remains as it is. Now the second expression has denominator. So we make it equal to lcm by multiplying both numerator and denominator by 2 into x minus 2 the whole. So this is equal to 2 into x plus 1 the whole into x minus 2 the whole and upon 2 into x plus 2 the whole into x minus 2 the whole. So now we will add the two expressions and this will be equal to x minus 3 over upon 2 into x minus 2 the whole into x plus 2 the whole plus. Now the second expression is 2 into x plus 1 the whole into x minus 2 the whole over upon 2 into x plus 2 the whole into x minus 2 the whole. Now this is equal to in the denominator we will have 2 into x minus 2 the whole into x plus 2 the whole and in the numerator we will have x minus 3 the whole plus 2 into x plus 1 the whole into x minus 2 the whole. And further this is equal to now in the numerator we will open the brackets and it will be x minus 3 plus 2 into now here a binomial is multiplied by a binomial. So this will be 2 into x into x minus 2 the whole plus 1 into x minus 2 the whole and upon 2 into x minus 2 the whole into x plus 2 the whole. And further this is equal to x minus 3 plus 2 into x square minus 2x plus x minus 2 the whole over upon 2 into x minus 2 the whole into x plus 2 the whole. Now this is equal to x minus 3 now here we will open the brackets so it will be plus 2x square minus 4x plus 2x minus 4 into x minus 2 the whole into x plus 2 the whole. And now combining the light terms in the numerator here you can see x and minus 4x are light terms minus 3 and minus 4 are light terms so this is equal to the whole plus of minus 3 minus 4 the whole over upon 2 into x minus 2 the whole into x plus 2 the whole and this is equal to now x minus 4x plus 2x is equal to minus x and here minus 3 minus 4 is minus 7 so it will be plus of minus 7 which is minus 7 plus whole upon 2 into x minus 2 the whole into x plus 2 we will write the expression in the numerator in descending order of powers of x so this will be equal to 2x square minus x minus 7 into x minus 2 the whole into x plus 2 the whole which is equal to 2x square minus x minus 7 x square minus 8 the required answer. Now let us start with the second part and the second part we have to find 3 upon x plus x upon 3x plus 4 now here we do not have any factors in the denominators of 2 expressions so here we simply take LCM as product of 2 denominators LCM of denominators is equal to x into 3x plus whole the whole. Now we need the denominators equivalent in both expressions using LCM we make it equal to LCM by multiplying both numerator and denominator by x plus 4 3 into 3x plus 4 the whole to the denominator the numerator and denominator by x equal to plus 4 the whole 3 upon x upon 3x plus 4 will be equal to the first expression is 3 into 3x plus 4 the whole upon x into 3x plus the whole plus upon x into 3x plus 4 the whole and further this is equal to now in denominator we will have x into 3x plus 4 the whole and in numerator we will have 3 into 3x plus 4 the whole plus x square in the numerator and this will be equal to 3 into 3x that is 9x plus 3 into 4 which is 12 plus m 3x plus 4 the whole. Now in the numerator we will write the expression in descending order of powers this will be equal to x square plus 9x plus 12 plus 4 the whole. This is the required answer for the second part hope you all have enjoyed the section.