To solve this problem, we need to find the second trinomial given the difference between two trinomials and one of the trinomials.
Given:
1. The difference between the two trinomials is [tex]\(5x^2 + 11x - 16\)[/tex].
2. One of the trinomials is [tex]\(3x^2 - 2x + 7\)[/tex].
Let's denote the trinomials as follows:
- [tex]\( T_1 = 3x^2 - 2x + 7 \)[/tex]
- [tex]\( D = 5x^2 + 11x - 16 \)[/tex] (this is the difference between the two trinomials)
We need to find a trinomial [tex]\( T_2 \)[/tex] such that:
[tex]\[ T_2 - T_1 = D \][/tex]
To find [tex]\( T_2 \)[/tex], we can rearrange the equation to solve for [tex]\( T_2 \)[/tex]:
[tex]\[ T_2 = D + T_1 \][/tex]
[tex]\[ T_2 = (5x^2 + 11x - 16) + (3x^2 - 2x + 7) \][/tex]
We will add the corresponding coefficients of [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constants from the two trinomials:
1. The coefficient of [tex]\( x^2 \)[/tex]:
[tex]\[ 5 + 3 = 8 \][/tex]
2. The coefficient of [tex]\( x \)[/tex]:
[tex]\[ 11 - 2 = 9 \][/tex]
3. The constant term:
[tex]\[ -16 + 7 = -9 \][/tex]
Thus, the resulting trinomial [tex]\( T_2 \)[/tex] is:
[tex]\[ T_2 = 8x^2 + 9x - 9 \][/tex]
Now, let's look at the given options:
A. [tex]\( 8x^2 + 9x - 9 \)[/tex]
B. [tex]\( 8x^2 + 13x - 23 \)[/tex]
C. [tex]\( 2x^2 + 9x - 9 \)[/tex]
D. [tex]\( 2x^2 + 13x - 23 \)[/tex]
The trinomial we calculated [tex]\( 8x^2 + 9x - 9 \)[/tex] matches option A.
Therefore, the correct answer is:
[tex]\[ \boxed{8x^2 + 9x - 9} \][/tex]
