 Hello students, let's work out the following problem. It says given that p is equal to x square minus x minus 6 q is equal to 3 upon x square plus x minus 2 r is equal to 4 upon x square minus 4x plus 3 Find p plus q divided by r. So let's now move on to the solution We have to find p plus q divided by r. So we first find p plus q Now p is 2 upon x square minus x minus 6 Plus q which is 3 upon x square plus x minus 2 Now to simplify this rational expression, we first factorize the denominators of p and q. So we have 2 upon x square minus 3x minus 2x minus x plus 3 upon x square plus 2x minus x minus 2 And this is again equal to 2 upon now taking x common from the first two terms We have x into x minus 3 and taking 2 common from the last two terms We have 2 into x minus 3 plus 3 upon again We take x common from the first two terms. We have x into x plus 2 now we take Minus 1 common from the last two terms. We have minus 1 into x plus 2 Now this is again equal to 2 upon Taking x minus 3 common. We have x minus 3 into x plus 2 plus 3 upon Taking x plus 2 common. We have x plus 2 into x minus 1. Now we take LCM LCM would be x minus 3 into x plus 2 into x minus 1 so in the numerator we have 2 into x minus 1 plus 3 into x minus 3 and this is again equal to 2 into x is 2x 2 into minus 1 is minus 2 plus 3 into x is 3x 3 into minus 3 is minus 9 upon x minus 3 into x plus 2 into x minus 1 So this is equal to 2x plus 3x is 5x minus 2 minus 9 is minus 11 upon x minus 3 into x plus 2 into x minus 1 So this is p plus q now we have to find p plus q divided by r p plus q divided by r Will be given by p plus q is this 5x minus 11 upon x minus 3 into x plus 2 into x minus 1 and r is 4 upon x square minus 4x plus 3 and this is again equal to 5x minus 11 upon x minus 3 into x plus 2 into x minus 1 Divided by r which is 4 upon now we factorize the denominator So we have x square minus 3x minus x plus 3 So this is again equal to 5x minus 11 upon x minus 3 into x plus 2 into x minus 1 divided by 4 upon Taking x common from the first two terms. We have x into x minus 3 Taking minus one common from the last two terms. We have x minus 1 into x minus 3 This is again equal to 5x minus 11 upon x minus 3 into x plus 2 into x minus 1 divided by 4 upon Taking x minus 3 common. We have x minus 3 into x minus 1 To simplify such rational expression you should always move stepwise now here you have to divide This rational expression by this and whenever you have something like a by b divided by C by D it becomes a by B multiplied by D by C you should always remember this that makes your calculations easy So this becomes equal to 5x minus 11 upon x minus 3 into x plus 2 into x minus 1 multiplied by x minus 3 Into x minus 1 upon 4. We have just taken the reciprocal of the second expression right Now here we see that x minus 3 gets cancelled with x minus 3 x minus 1 gets cancelled with x minus 1 and we are left with 5x minus 11 upon x plus 2 into 1 by 4 so this is equal to 5x minus 11 upon 4 into x plus 2 and this is the required answer So this completes the session and the question life for now take care. Have a good day