To find the other trinomial, we need to determine what trinomial added to [tex]\(4x^2 + 3x - 2\)[/tex] results in [tex]\(6x^2 - 5x + 4\)[/tex].
Let's denote the other trinomial as [tex]\(Ax^2 + Bx + C\)[/tex].
We start from the equation:
[tex]\[ (4x^2 + 3x - 2) + (Ax^2 + Bx + C) = 6x^2 - 5x + 4 \][/tex]
To find the coefficients [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex], we equate the coefficients of [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant terms on both sides of the equation.
1. Coefficient of [tex]\(x^2\)[/tex]:
[tex]\[ 4 + A = 6 \][/tex]
Solving for [tex]\(A\)[/tex]:
[tex]\[ A = 6 - 4 \][/tex]
[tex]\[ A = 2 \][/tex]
2. Coefficient of [tex]\(x\)[/tex]:
[tex]\[ 3 + B = -5 \][/tex]
Solving for [tex]\(B\)[/tex]:
[tex]\[ B = -5 - 3 \][/tex]
[tex]\[ B = -8 \][/tex]
3. Constant term:
[tex]\[ -2 + C = 4 \][/tex]
Solving for [tex]\(C\)[/tex]:
[tex]\[ C = 4 + 2 \][/tex]
[tex]\[ C = 6 \][/tex]
Therefore, the other trinomial is:
[tex]\[ 2x^2 - 8x + 6 \][/tex]
Thus, the correct answer is:
[tex]\[ 2x^2 - 8x + 6 \][/tex]
So, the correct option is [tex]\( \boxed{2x^2 - 8x + 6} \)[/tex].
