Let's solve the problem step by step:
We are given two polynomials:
[tex]\[ P(x) = 11x^2 - 5 \][/tex]
[tex]\[ Q(x) = x + 4 \][/tex]
We want to find the sum of these polynomials, [tex]\( P(x) + Q(x) \)[/tex].
1. Identify the coefficients of each polynomial:
For [tex]\( P(x) = 11x^2 - 5 \)[/tex]:
- The coefficient of [tex]\( x^2 \)[/tex] is 11.
- The coefficient of [tex]\( x \)[/tex] is 0 (since there is no [tex]\( x \)[/tex] term).
- The constant term is -5.
For [tex]\( Q(x) = x + 4 \)[/tex]:
- The coefficient of [tex]\( x^2 \)[/tex] is 0 (since there is no [tex]\( x^2 \)[/tex] term).
- The coefficient of [tex]\( x \)[/tex] is 1.
- The constant term is 4.
2. Add the corresponding coefficients:
For [tex]\( x^2 \)[/tex] terms:
[tex]\( 11 + 0 = 11 \)[/tex]
For [tex]\( x \)[/tex] terms:
[tex]\( 0 + 1 = 1 \)[/tex]
For the constant terms:
[tex]\( -5 + 4 = -1 \)[/tex]
3. Write out the sum polynomial using the summed coefficients:
Combining these results, we get:
[tex]\[
P(x) + Q(x) = 11x^2 + x - 1
\][/tex]
So, the sum of the given polynomials is:
[tex]\[ 11x^2 + x - 1 \][/tex]
Thus, the option that corresponds to this polynomial is:
[tex]\[ \boxed{11x^2 + x - 1} \][/tex]
