Question 31·Easy·Equivalent Expressions

When you subtract $(x - 1)(2x + 5)$, think of it as subtracting each term in that expanded trinomial. How does the minus sign affect each term?

After distributing the minus sign, group $x^2$ terms together, $x$ terms together, and constants together, and then combine each group.

In one Desmos line, type the full expression: (4x - 3)(x + 2) - (x - 1)(2x + 5) and let Desmos simplify it; note the simplified form that appears on the right.

On separate lines, enter each choice, for example A(x) = 2x^2 + 8x - 1, B(x) = 6x^2 + 2x - 1, etc., and compare their simplified forms with the simplified form of the original expression. The correct choice is the one whose expression exactly matches the simplified original expression for all $x$.

First, expand $(4x - 3)(x + 2)$.

• Multiply term by term (FOIL):
  • First: $4x \cdot x = 4x^2$
  • Outer: $4x \cdot 2 = 8x$
  • Inner: $-3 \cdot x = -3x$
  • Last: $-3 \cdot 2 = -6$

Combine like terms:

Now expand $(x - 1)(2x + 5)$:

• First: $x \cdot 2x = 2x^2$
• Outer: $x \cdot 5 = 5x$
• Inner: $-1 \cdot 2x = -2x$
• Last: $-1 \cdot 5 = -5$

Combine like terms:

Substitute the expanded forms back into the original expression:

becomes

Now the problem is to simplify this difference of two trinomials.

Distribute the minus sign to every term in the second parentheses:

Now combine like terms:

• $x^2$ terms:

• $x$ terms:

• Constant terms:

So the simplified expression is:

which matches answer choice D.