Question

Simplify:
$(pq-qr+pr)(pq+qr)-(pr+pq)(p+q-r)$

Answer 1

Given $\begin{array}{c}\left(pq-qr+pr\right)\left(pq+qr\right)-\left(pr+pq\right)\left(p+q-r\right)\end{array}$
$=pq(pq+qr)-qr(pq+qr)+pr(pq+qr)-pr(p+q-r)-pq(p+q-r)$
$\begin{array}{c}=\left[p^2{q}^2+p{q}^{2}r-p{q}^{2}r-q^2{r}^2+p^{2}qr+pq{r}^2\right]-\left[p^{2}r+pqr-p{r}^2+p^{2}q+p{q}^2-pqr\right]\\ =p^2{q}^2+p{q}^{2}r-p{q}^{2}r-q^2{r}^2+p^{2}qr+pq{r}^2-p^{2}r-pqr+p{r}^2-p^{2}q-p{q}^2+pqr\\ =p^2{q}^2-q^2{r}^2+p^{2}qr+pq{r}^2-p^{2}r+p{r}^2-p^{2}q-p{q}^2\end{array}$