Question 62·Medium·Equivalent Expressions

When you factor out $(x^2 + 4x)$, you will get something like $[(2x - 3) - (x - 5)](x^2 + 4x)$. Be careful to distribute the minus sign to both terms inside $(x - 5)$.

After simplifying the expression inside the brackets, multiply the resulting binomial by $(x^2 + 4x)$ and then combine like terms to get a single polynomial.

In Desmos, type f(x) = (2x - 3)(x^2 + 4x) - (x - 5)(x^2 + 4x) to represent the given expression as a function.

On new lines, type g(x) = 3x^3 + 4x^2 + 8x, h(x) = x^3 + 6x^2 - 8x, p(x) = x^3 + 8x, and q(x) = x^3 + 6x^2 + 8x.

Either:

• Look at the graphs: the correct choice will have a graph that lies exactly on top of the graph of $f(x)$ for all visible $x$-values, or
• Open a table for $f(x)$ and each choice (click the gear icon, then “Table”) and compare outputs at several $x$-values (like $x = -2, -1, 0, 1, 2$). The correct answer will match $f(x)$ at every tested value.

Notice that both terms contain $(x^2 + 4x)$:

Use the distributive property to factor out $(x^2 + 4x)$:

Now you only need to simplify the expression inside the brackets and then multiply by $(x^2 + 4x)$.

Simplify $[(2x - 3) - (x - 5)]$.

First distribute the negative sign across $(x - 5)$:

Now combine like terms:

• Combine $2x$ and $-x$ to get $x$.
• Combine $-3$ and $5$ to get $2$.

So the bracket becomes $x + 2$, and the whole expression is

Now multiply $(x + 2)(x^2 + 4x)$ using distribution:

• Multiply $x$ by each term in $(x^2 + 4x)$:
  • $x \cdot x^2 = x^3$
  • $x \cdot 4x = 4x^2$
• Multiply $2$ by each term in $(x^2 + 4x)$:
  • $2 \cdot x^2 = 2x^2$
  • $2 \cdot 4x = 8x$

So you get four terms:

Combine the like terms $4x^2$ and $2x^2$:

So the simplified expression is

which corresponds to answer choice D.