Question

question 2 of 6
select the correct answer.
the difference of two trinomials is $x^2 - 10x + 2$. if one of the trinomials is $3x^2 - 11x - 4$, then which expression could be the other trinomial?
$2x^2 - x - 2$
$2x^2 + x + 6$
$4x^2 + 21x + 6$
$4x^2 - 21x - 2$

Explanation:

Step1: Let the other trinomial be \( A \). We know that the difference of the two trinomials is \( x^2 - 10x + 2 \), and one trinomial is \( 3x^2 - 11x - 4 \). There are two cases: either \( (3x^2 - 11x - 4) - A = x^2 - 10x + 2 \) or \( A - (3x^2 - 11x - 4) = x^2 - 10x + 2 \). We will check both cases, but first let's solve the first case for \( A \):

Starting with \( (3x^2 - 11x - 4) - A = x^2 - 10x + 2 \), we can rearrange to get \( A = (3x^2 - 11x - 4) - (x^2 - 10x + 2) \).

Step2: Simplify the expression for \( A \):

\[

$$\begin{align*} A&= 3x^2 - 11x - 4 - x^2 + 10x - 2\\ &= (3x^2 - x^2) + (-11x + 10x) + (-4 - 2)\\ &= 2x^2 - x - 6 \end{align*}$$

\]
This is not one of the options, so let's check the second case: \( A - (3x^2 - 11x - 4) = x^2 - 10x + 2 \). Rearranging to solve for \( A \), we get \( A = (x^2 - 10x + 2) + (3x^2 - 11x - 4) \).

Step3: Simplify this expression for \( A \):

\[

$$\begin{align*} A&= x^2 - 10x + 2 + 3x^2 - 11x - 4\\ &= (x^2 + 3x^2) + (-10x - 11x) + (2 - 4)\\ &= 4x^2 - 21x - 2 \end{align*}$$

\]
This is one of the options (the fourth option). Let's verify with the first case's result not matching and the second case's result matching option D. Wait, but let's also check by subtracting the given trinomial from the option to see if we get the difference. Let's take option D: \( 4x^2 - 21x - 2 \) and subtract \( 3x^2 - 11x - 4 \):
\[

$$\begin{align*} (4x^2 - 21x - 2) - (3x^2 - 11x - 4)&= 4x^2 - 21x - 2 - 3x^2 + 11x + 4\\ &= (4x^2 - 3x^2) + (-21x + 11x) + (-2 + 4)\\ &= x^2 - 10x + 2 \end{align*}$$

\]
Which matches the given difference. Let's also check option A: \( 2x^2 - x - 2 \) minus \( 3x^2 - 11x - 4 \) is \( -x^2 + 10x + 2 \), not the difference. Option B: \( 2x^2 + x + 6 \) minus \( 3x^2 - 11x - 4 \) is \( -x^2 + 12x + 10 \), not matching. Option C: \( 4x^2 + 21x + 6 \) minus \( 3x^2 - 11x - 4 \) is \( x^2 + 32x + 10 \), not matching. So the correct one is the fourth option.

Answer:

D. \( 4x^2 - 21x - 2 \)