Multiply $x$ by each term in $(1 + x + x^2 + x^3)$, then multiply $-1$ by each term in the same parentheses. Write all resulting terms out.

After expansion, collect like terms (terms with the same power of $x$). Only after that, remember to include the $+1$ at the end and see which terms cancel.

Type f(x) = (x - 1)(1 + x + x^2 + x^3) + 1 into Desmos. This is the function you want to match.

On new lines, enter g(x) = x^4, h(x) = x^4 - 1, j(x) = x^3, and k(x) = 1. Look at the graphs to see which one lies exactly on top of the graph of $f(x)$ for all visible $x$-values.

Click the gear icon next to $f(x)$ and choose "Convert to table" (or add a table manually) to create $x$-values. Then, in separate columns, evaluate the answer choice functions at the same $x$-values. The correct choice will have matching $y$-values with $f(x)$ for every tested $x$.

We want an equivalent, simplified expression for

The usual way is to expand the product using the distributive property and then combine like terms.

First, distribute $x$ across the parentheses:

Next, distribute $-1$ across the same parentheses:

So the product part becomes

Now include the $+1$ at the end of the original expression.

Write the full expanded expression including the $+1$:

Group like terms:

• $x$ and $-x$ cancel.
• $x^2$ and $-x^2$ cancel.
• $x^3$ and $-x^3$ cancel.
• $-1$ and $+1$ cancel.

The only term that does not cancel is $x^4$, so the entire expression simplifies to $x^4$, which corresponds to choice A.