5. Mr. Silva makes hover boards at his manufacturing plant. From 2000 through 2010 , \( \begin{array}{l}N(x)=x^{2}+10 x+150 \quad \text { is the number of units produced } \\ C(x)=-x^{2}+x+125\end{array} \quad \) is the average cost per unit (in dollars) \( x \) is the number of years since 2000 . a. Write a polynomial \( T(x) \) that can be used to model Mr. Silva's total manufacturing cost for those years.

To find Mr. Silva's total manufacturing cost \( T(x) \), we can multiply the number of units produced \( N(x) \) by the average cost per unit \( C(x) \). This gives us the polynomial: \[ T(x) = N(x) \times C(x) = (x^2 + 10x + 150)(-x^2 + x + 125) \] Now, to get the complete expression for \( T(x) \), we would expand this product, which will yield a polynomial representing Mr. Silva's total manufacturing cost from 2000 to 2010. To visualize this total manufacturing cost, think of it as Mr. Silva's treasure chest where the amount of golden coins he has increases with every year of production, but the treasure’s worth is also influenced by the market costs of producing those hoverboards. This can help him make better decisions on how to scale his business! And did you know? The hoverboard became widely popular in 2015, largely due to media coverage and viral videos. However, the concept dates back to the '80s with the "hoverboard" seen in movies like "Back to the Future Part II." It’s interesting how innovation can take years to really catch on!