 Hello and welcome to the session. In this session, we will discuss a question which says that find x upon x square minus 1x plus 3 by whole minus of x minus 3 home upon x square plus 2x minus 3 by whole and write the obtained single reflection in its lowest terms. Now, let us start with the solution of the given question. Now, here we have to subtract these two rational expressions. Now, here first of all we will find a form of the denominators of these two fractions. Now, for this we will factorize the denominators of the two fractions. Now, denominator of first fraction is x square minus 4x plus 3. Now, let us factorize it. Now, for factorizing this expression, we will split the middle term into two terms such that some of those two terms is equal to the middle term that is minus 4x and multiplication of coefficients of those two terms is equal to product of constant term efficient of x square that is 3 into 1 which is equal to 3. So, this will be equal to x square minus x minus 3x plus 3 at minus x and minus 3x it will give us minus 4x and if we multiply minus 1 and minus 3 it will give us plus 3. Further, this is equal to the first two terms taking x common it will be x into x minus 1 the whole and from last two terms taking minus 3 common it will be minus 3 into x minus 1 the whole. Now, from this complete expression taking x minus 1 common it will be x minus 1 the whole into x minus 3 the whole. So, on factorizing the denominator of first fraction we obtain x minus 1 the whole into x minus 3 the whole. Now, the denominator of second fraction is x minus 3. Now, here for factorizing this expression we will use the same method which we have discussed earlier. Now, this is equal to x minus 3 which is equal to now taking x common from first two terms this is x into x plus 3 the whole. Common from last two terms it will be minus 1 into x plus 3 the whole which is equal to x plus 3 the whole into x minus 1 the whole. So, on factorizing the denominator of second fraction we obtain x plus 3 the whole into x minus 1 the whole. And now we have to find lcm of denominators. Now, lcm of denominators is the product in one or both the denominators and here the repeated factors are taken only once denominators will be equal to. Now, here you can see that x minus 1 is the repeated factor it will be taken only once. So, lcm will be equal to x minus 3 the whole. Now, using the lcm we will make the denominators in both the expressions or we can say we will make the denominators in both these fractions. Now, the first factor is 4x plus 3 can be written as upon this denominator is equal to x minus 1 the whole into x minus 3 the whole. Now, this first expression that is this rational expression has denominator x minus 1 the whole into x minus 3 the whole lcm by multiplying both numerator and denominator by 3 the whole whole upon x minus 1 the whole into x minus 3 the whole into x plus 3 the whole. Now, we can say the second rational expression is x minus 3 over x square plus 2x minus 3 which can be written as now on factorizing denominator of this second fraction that is x square plus 2x minus 3 is equal to minus 1 the whole into x plus 3 the whole for making denominator equal to lcm. We multiply both numerator and denominator by x minus 3 the whole whole into x minus 3 the whole minus 1 the whole into x plus 3 the whole into x minus 3 the whole. Now, using these values, let us subtract these two rational expressions. So, this will be equal to x plus 3 the whole whole upon x minus 1 the whole into x minus 3 the whole into x plus 3 the whole minus the whole into x minus 3 the whole whole upon minus 1 the whole into x minus 3 the whole into x plus 3 the whole. And this complete whole now subtracting we get in the numerator we will have x into x plus 3 the whole minus of x minus 3 the whole into x minus 3 the whole and this complete whole. For example, x minus 1 the whole into x minus 3 the whole into x plus 3 the whole 3x minus into x minus 3 the whole minus 3 into x minus 3 the whole and this complete whole whole into x minus 3 the whole into x plus 3 the whole. And this is equal to x square plus 3x minus of x square minus 3x. Now minus 3 into x is minus 3x and minus 3 into minus 3 is plus 9. Now minus 1 by whole into x minus 3 by whole into x plus 3 by whole. Now this is equal to x square minus 3x minus 3x is minus 6x plus 9 by whole into x minus 3 by whole into x plus 3 by whole. Now again, let us look at the brackets in the numerator. This is just 3x. Now minus of minus 6x is plus 6x and minus of plus 9 is minus 9. Holocon x minus 1 by whole into x minus 3 by whole into x plus 3 by whole. Now combining the right terms x square minus x square by whole plus of 3x plus 6x by whole minus 9. Holocon x minus 1 by whole into x minus 3 by whole into x plus 3 by whole. x square minus x square is 0. 3x plus 6x is 9x minus 9. This is 1 by whole into x minus 3 by whole into x plus 3 by whole. And now we have to write in its lowest terms as the greatest common factor in numerator. So taking 9 common from both the terms in numerator this will be 9 into x minus 1 by whole minus 1 by whole into x minus 3 by whole into x plus 3 by whole. And now we will consider the common factor is 1 by whole in both numerator and denominator and this will be equal to 9 upon 1 into x plus 3 by whole. So I am subtracting these two fractions and we can say I am subtracting these two rational expressions. We obtain 9 upon x minus 3 by whole into x plus 3 by whole. So this is the required answer and this completes our session. Hope you all have enjoyed the session.