Question 186·Medium·Advanced Math

Choose $x=2$ to keep the products small. Type $x_1=2$ in Desmos, then type the original expression with each $x$ changed to $x_1$ so Desmos evaluates it. Desmos prints $48$, so an equivalent expression must give $48$ at $x=2$.

Type the first three choices below the original, again using $x_1$. Desmos prints $56$, $16$, and $24$, respectively. None equals $48$, so none of those expressions can be equivalent: they already disagree at $x=2$.

Type $x_1^3+6x_1^2+8x_1$. Desmos prints $48$, matching the original. One matching input alone would not prove equivalence, but the other three choices have failed at this input. The remaining choice is $x^3+6x^2+8x$. Choice D.

Both terms contain the same factor, $x^2+4x$. The distributive property works backward to pull it out. Keep the subtraction between the factors left behind. Factor it out:

The minus before $(x-5)$ means multiply the whole factor by $-1$. Distribute that minus sign:

Combine the $x$ terms and the numbers inside the brackets:

The distributive property multiplies each term in $x^2+4x$ by both $x$ and $2$. Write all four products:

The like terms $2x^2$ and $4x^2$ have the same power of $x$, so add them:

This gives the equivalent expression for every $x$, not only the input tested in Desmos. Choice D.