 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that solve for x and identify the extraneous solution if any. And the equation given is x plus 3 whole upon x square plus 5x plus 4 minus 4 upon x square minus 4x minus 32 is equal to minus 3 upon x square minus 7x minus 8. Now let us start with the solution of the given question. Here we are given this rational equation and we have to solve it for x. First of all we will factorize the quadratic expressions given in the denominator. Now here we get x square minus 5x plus 4 after factorization can be written as x plus 1 the whole into x plus 4 the whole. Similarly x square minus 4x minus 32 after factorization can be written as x plus 4 the whole into x minus 8 the whole and x square minus 7x minus 8 on factorization becomes x plus 1 the whole into x minus 8 the whole. And therefore we can write this equation as x plus 3 whole upon x plus 1 the whole into x plus 4 the whole minus 4 upon x minus 8 the whole into x plus 4 the whole is equal to minus 3 upon x minus 8 the whole into x plus 1 the whole. Now here least common denominator of all the denominators of this equation is given by x plus 1 the whole into x plus 4 the whole into x minus 8 the whole. So we multiply both sides of the equation by the least common denominator that is x plus 1 the whole into x plus 4 the whole into x minus 8 the whole. We get x plus 1 the whole into x plus 4 the whole into x minus 8 the whole into x plus 3 whole upon x plus 1 the whole into x plus 4 the whole minus x plus 1 the whole into x plus 4 the whole into x minus 8 the whole into 4 upon x minus 8 the whole into x plus 4 the whole and this is equal to x plus 1 the whole into x plus 4 the whole into x minus 8 the whole into minus 3 upon x minus 8 the whole into x plus 1 the whole. Now this further implies that x minus 8 the whole into x plus 3 the whole minus 4 into x plus 1 the whole is equal to minus 3 into x plus 4 the whole. Now opening the brackets here we get x into x that is x square plus x into 3 that is plus 3x minus 8 into x is minus 8x minus 8 into 3 is minus 24 minus 4 into x is minus 4x minus 4 into 1 is minus 4 and this is equal to minus 3 into x is minus 3x minus 3 into 4 is minus 12 now simply find further we get x square now plus 3x minus 8x minus 4x is minus 9x now minus 24 minus 4 is minus 28 and this is equal to minus 3x minus 12. Which further implies that x square minus 9x minus 28 plus 3x plus 12 is equal to 0 which further implies that x square now minus 9x plus 3x is minus 6x and minus 28 plus 12 is minus 16 and this is equal to 0. Now on factorization x square minus 6x minus 16 can be written as x plus 2 the whole into x minus 8 the whole. So here we have x plus 2 the whole into x minus 8 the whole is equal to 0 which implies that x is equal to minus 2 and 8. Now we have two solutions of the equation and these are x is equal to 8 and x is equal to minus 2 and we know that here least common denominator is given by x plus 1 the whole into x plus 4 the whole into x minus 8 the whole and here x is equal to 8 will make the least common denominator equal to 0 that is if we put the value of x as 8 in this expression its value will become 0. Now we know that a rational equation is undefined at that value where least common denominator is equal to 0 and if in the solution set we get a value where one of the expressions given in the equation becomes undefined then we omit that value from the answer set and that value is called extraneous solution so here the equation will become undefined at x is equal to 8 and it is an extraneous solution and we will omit it from solution set. Thus x is equal to minus 2 is the only solution of the given rational equation this is the required answer this completes our session hope you enjoyed this session.