Select the correct answer.
The difference between two trinomials is $5 x^{2}+11 x-16$. If one of the trinomials is $3 x^{2}-2 x+7$, then which expression could be the other trinomial?
A. $2 x^{2}+13 x-23$
B. $2 x^{2}+9 x-9$
C. $8 x^{2}+9 x-9$
D. $8 x^{2}+13 x-23$
Answer
\(\boxed{8x^2 + 9x - 9}\) is the other trinomial.
Step 1 ：Given the difference between two trinomials is $5x^2 + 11x - 16$. One of the trinomials is $3x^2 - 2x + 7$. Let the other trinomial be $ax^2 + bx + c$.
Step 2 ：Set up the equation: $(ax^2 + bx + c) - (3x^2 - 2x + 7) = 5x^2 + 11x - 16$
Step 3 ：Simplify the equation: $(a - 3)x^2 + (b + 2)x + (c - 7) = 5x^2 + 11x - 16$
Step 4 ：Equate the coefficients of the corresponding terms:
Step 5 ：$a - 3 = 5$
Step 6 ：$b + 2 = 11$
Step 7 ：$c - 7 = -16$
Step 8 ：Solve for $a$, $b$, and $c$: $a = 8$, $b = 9$, $c = -9$
Step 9 ：\(\boxed{8x^2 + 9x - 9}\) is the other trinomial.
