To find the sum of two polynomials, we add the coefficients of the like terms. Let's examine the given polynomials and add their coefficients step-by-step.
The given polynomials are:
[tex]\[ (3n^2 - 5n + 6) \][/tex]
and
[tex]\[ (-8n^2 - 3n - 2) \][/tex]
We will add the coefficients of like terms (terms involving [tex]\( n^2 \)[/tex], [tex]\( n \)[/tex], and the constant terms) to get the resulting polynomial.
1. Add the coefficients of the [tex]\( n^2 \)[/tex] terms:
The coefficient of [tex]\( n^2 \)[/tex] in the first polynomial is 3.
The coefficient of [tex]\( n^2 \)[/tex] in the second polynomial is -8.
Adding these coefficients:
[tex]\[ 3 + (-8) = -5 \][/tex]
So, the coefficient of [tex]\( n^2 \)[/tex] in the resulting polynomial is -5.
2. Add the coefficients of the [tex]\( n \)[/tex] terms:
The coefficient of [tex]\( n \)[/tex] in the first polynomial is -5.
The coefficient of [tex]\( n \)[/tex] in the second polynomial is -3.
Adding these coefficients:
[tex]\[ -5 + (-3) = -8 \][/tex]
So, the coefficient of [tex]\( n \)[/tex] in the resulting polynomial is -8.
3. Add the constant terms:
The constant term in the first polynomial is 6.
The constant term in the second polynomial is -2.
Adding these constants:
[tex]\[ 6 + (-2) = 4 \][/tex]
So, the constant term in the resulting polynomial is 4.
Combining these results, we get the resulting polynomial:
[tex]\[ -5n^2 - 8n + 4 \][/tex]
Thus, the sum of the polynomials [tex]\( \left(3n^2 - 5n + 6\right) \)[/tex] and [tex]\( \left(-8n^2 - 3n - 2\right) \)[/tex] is:
[tex]\[ \boxed{-5n^2 - 8n + 4} \][/tex]
Among the given choices, the correct answer is:
[tex]\[ -5n^2 - 8n + 4 \][/tex]
So the correct choice is:
[tex]\[ \boxed{-5n^2 - 8n + 4} \][/tex]
