Question 93·Easy·Advanced Math

Type $y=(4x-3)(x+2)-(x-1)(2x+5)$ on line 1. Desmos draws a parabola, a U-shaped curve. Keep the minus sign before the whole second product so the graph represents the expression in the question.

On line 2, type $y=2x^2+8x-1$. Desmos shows that this curve separates from line 1 across the window. A shared point isn't enough: equivalent expressions must agree for every $x$.

On line 3, type $y=6x^2+2x-1$. Desmos shows that this curve also separates from line 1 across the window. Its intersection with the original curve doesn't make the expressions equivalent.

On line 4, type $y=2x^2+2x+1$. Desmos shows that its curve stays above the original rather than overlapping it. So this choice isn't an equivalent expression.

Type $y=2x^2+2x-1$ on line 5. Desmos shows full overlap with the original curve. Hide and show line 5: the curve doesn't move, so $2x^2+2x-1$ is equivalent to the given expression. Choice D.

Use the distributive property: multiply each term in $4x-3$ by each term in $x+2$. Write all four products before combining them: $(4x-3)(x+2)=4x^2+8x-3x-6$.

The $8x$ and $-3x$ are like terms: they have the same variable part. Combine them: $4x^2+8x-3x-6=4x^2+5x-6$. Keep the $-6$ for the final subtraction.

Multiply each term in $x-1$ by each term in $2x+5$. Include $-1$ times both terms: $(x-1)(2x+5)=2x^2+5x-2x-5$.

Combine the like terms $5x$ and $-2x$: $2x^2+5x-2x-5=2x^2+3x-5$. This is the whole product that the original expression subtracts.

The original minus sign applies to every term in $2x^2+3x-5$. Subtract it from the first product, changing all three signs: $(4x^2+5x-6)-(2x^2+3x-5)=4x^2+5x-6-2x^2-3x+5$. In particular, subtracting $-5$ gives $+5$.

Combine the $x^2$ terms, the $x$ terms, and the constants separately: $4x^2+5x-6-2x^2-3x+5=2x^2+2x-1$. The equivalent expression is $2x^2+2x-1$. Choice D.