To find the other trinomial, we need to determine a trinomial that, when subtracted from the given trinomial [tex]\(3x^2 - 2x + 7\)[/tex], results in the difference [tex]\(5x^2 + 11x - 16\)[/tex].
Let's denote the other trinomial as [tex]\(Ax^2 + Bx + C\)[/tex]. According to the problem, the equation we are dealing with is:
[tex]\[
(Ax^2 + Bx + C) - (3x^2 - 2x + 7) = 5x^2 + 11x - 16
\][/tex]
We can rewrite this equation to separate the coefficients:
1. For the [tex]\(x^2\)[/tex] term:
[tex]\[
A - 3 = 5
\][/tex]
2. For the [tex]\(x\)[/tex] term:
[tex]\[
B - (-2) = 11 \implies B + 2 = 11
\][/tex]
3. For the constant term:
[tex]\[
C - 7 = -16
\][/tex]
Now we solve these equations one by one:
1. For the [tex]\(x^2\)[/tex] term:
[tex]\[
A - 3 = 5 \implies A = 5 + 3 = 8
\][/tex]
2. For the [tex]\(x\)[/tex] term:
[tex]\[
B + 2 = 11 \implies B = 11 - 2 = 9
\][/tex]
3. For the constant term:
[tex]\[
C - 7 = -16 \implies C = -16 + 7 = -9
\][/tex]
Thus, the other trinomial is:
[tex]\[
8x^2 + 9x - 9
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{8x^2 + 9x - 9}
\][/tex]
Option B is correct.
