Simplify the following:

(pq-qr + pr)(pq + qr)−(pr + pq)(p + q−r)

We have , (pq-qr + pr)(pq + qr)−(pr + pq)(p + q−r)

(pq-qr + pr)(pq + qr) = (pq + qr)( pq-qr + pr) (using Commutative law)

= pq(pq-qr + pr) + qr( pq-qr + pr) (using distributive law )

= p²q²-pq²r + p²qr + pq²r-q²r² + pqr² = p²q² + p²qr + pqr²-q²r²

(pr + pq)(p + q−r) = pr(p + q-r) + pq(p + q−r) (using distributive law )

= p²r + prq-pr² + p²q + pq²-prq

= p²r-pr² + p²q + pq²

(pq-qr + pr)(pq + qr)−(pr + pq)(p + q−r) = p²q² + p²qr + pqr²-q²r²-( p²r-pr² + p²q + pq²)

= p²q² + p²qr + pqr²-q²r²-p²r + pr²-p²q-pq²

Hence, (pq-qr + pr)(pq + qr)−(pr + pq)(p + q−r) = p²q² + p²qr + pqr²-q²r²-p²r + pr²-p²q-pq²