After distributing, write down every term from both products in one long sum so you can see all the like terms.

Compare terms with the same power of $r$. Do any come in opposite-sign pairs, like $+r^2$ and $-r^2$, that cancel each other out?

In Desmos, type f(r) = (1 + r)(1 - r + r^2 - r^3 + r^4) to define the original expression as a function of $r$.

Type each option as its own function, for example g(r) = 1 - r^5, h(r) = 1 + r^5, p(r) = 1 - r^6, and q(r) = 1 + r^6.

Either:

• Turn on a table for f and each other function (click the gear icon and select “Table”) and compare their $y$-values for several $r$-values, or
• Look at the graphs and see which function’s graph lies exactly on top of the graph of f(r).

The option whose function always matches f(r) is the equivalent expression.

Apply the distributive property: multiply the entire second parentheses by 1, and then by $r$.

• Multiplying by 1 leaves the expression the same:

• Multiplying by $r$ increases the power of $r$ in each term by 1:

Now add the two results together:

Group the like terms (same power of $r$):

• Constant term: $1$
• $r$ terms: $-r$ and $+r$
• $r^2$ terms: $+r^2$ and $-r^2$
• $r^3$ terms: $-r^3$ and $+r^3$
• $r^4$ terms: $+r^4$ and $-r^4$
• $r^5$ term: $+r^5$

Combine each pair of like terms:

• $-r + r = 0$
• $r^2 - r^2 = 0$
• $-r^3 + r^3 = 0$
• $r^4 - r^4 = 0$

All the intermediate powers of $r$ cancel out, leaving just the constant term and the $r^5$ term:

So the equivalent expression is $1 + r^5$.