To determine the total profit from the sale of both items, we need to sum the polynomial expressions representing the profit for each item. The profit polynomial for the first item is:
[tex]\[ P_1(s) = s^3 - 70s^2 + 1500s - 10800 \][/tex]
The profit polynomial for the second item is:
[tex]\[ P_2(s) = s^3 - 30s^2 + 450s - 5000 \][/tex]
To find the total profit, we sum these two polynomials:
[tex]\[ P_{\text{total}}(s) = P_1(s) + P_2(s) \][/tex]
This means we need to add the corresponding coefficients of each term in the polynomials.
Let's do this step-by-step:
1. Cubic terms:
[tex]\[ s^3 + s^3 = 2s^3 \][/tex]
2. Quadratic terms:
[tex]\[ -70s^2 - 30s^2 = -100s^2 \][/tex]
3. Linear terms:
[tex]\[ 1500s + 450s = 1950s \][/tex]
4. Constant terms:
[tex]\[ -10800 - 5000 = -15800 \][/tex]
Combining all these results, the polynomial that expresses the total profit from the sale of both items is:
[tex]\[ P_{\text{total}}(s) = 2s^3 - 100s^2 + 1950s - 15800 \][/tex]
Therefore, the correct polynomial is:
[tex]\[ 2s^3 - 100s^2 + 1950s - 15800 \][/tex]
The correct answer is:
[tex]\[ \boxed{2s^3 - 100s^2 + 1950s - 15800} \][/tex]
