Expand and simplify (p - 1)(q - 1) + 1.

Step-by-step solution

1. 1

   Expand the product first, then deal with the loose term. The $+1$ sits outside the brackets, so it must not be folded in until the multiplication is finished. Order of operations puts the product before the addition.

2. 2

   Multiply the two binomials. Each term of the first bracket meets each term of the second — four products in all:

   $(p-1)(q-1)=p\cdot q+p\cdot(-1)+(-1)\cdot q+(-1)(-1)=pq-p-q+1$

   The final $+1$ comes from $(-1)(-1)$; a negative times a negative is positive.

3. 3

   Note that no terms combine. $p$ and $q$ are different variables, so $-p$ and $-q$ are not like terms and cannot be merged into $-2p$ or $-pq$. The expansion has four distinct terms.

4. 4

   Add the outside 1.

   $pq-p-q+1+1=pq-p-q+2$

   The two constants add rather than cancel, which is the only real subtlety in the problem.

5. 5

   Verify with random substitutions. Comparing the original with $pq-p-q+2$ at $50$ random pairs $(p,q)$ drawn from $[-8,8]$ gives agreement to machine precision at every point ✓. Spot check at $p=4,q=7$: $(3)(6)+1=19$, and $28-4-7+2=19$ ✓.