Write and simplify a polynomial expression that represents the length of the south side of the field. 𝟓𝒂 − 𝟑 𝒂 𝟐 − 𝟖 𝒂 𝟐 − 𝟓𝒂 + 𝟏� ####

1. ### Break Down the Problem

To solve this problem, we need to simplify the polynomial expression given: $5a - 3 + a^2 - 8a^2 - 5a + 1$

2. ### Relevant Concepts

To simplify the polynomial, we will combine like terms. This involves identifying and grouping terms that have the same degree.

3. ### Analysis and Detail

Let's organize the expression:

• Identify the terms by their degree:
  • Constant terms: $-3$ and $1$
  • Linear terms: $5a$ and $-5a$
  • Quadratic terms: $a^2$ and $-8a^2$

Now we can combine the like terms:

• Constant terms: $-3 + 1 = -2$

• Linear terms: $5a - 5a = 0$

• Quadratic terms: $a^2 - 8a^2 = -7a^2$

Putting it all together, we have: $-7a^2 + 0 - 2 = -7a^2 - 2$

4. ### Verify and Summarize

We have combined all like terms correctly and summarized the polynomial expression.

Final Answer

The simplified polynomial expression that represents the length of the south side of the field is: $-7a^2 - 2$