To simplify the given polynomial \(2 + 7c - 4c^2 - 3c + 4\) and write it in standard form, follow these steps:
1. Identify and combine like terms:
- Constant terms: \(2\) and \(4\)
- Linear terms (terms with \(c\)): \(7c\) and \(-3c\)
- Quadratic term (term with \(c^2\)): \(-4c^2\)
2. Combine the constant terms:
[tex]\[
2 + 4 = 6
\][/tex]
3. Combine the linear terms:
[tex]\[
7c - 3c = 4c
\][/tex]
4. Write down the quadratic term:
[tex]\[
-4c^2
\][/tex]
5. Put all the simplified terms together to write the polynomial in standard form:
[tex]\[
-4c^2 + 4c + 6
\][/tex]
Now, we need to match this simplified polynomial to one of the given options.
The given options are:
1. \(-4c^2 + 4c + 6\)
2. \(-7c^2 + 10c + 6\)
3. \(-4c^2 + 4c + 8\)
4. \(-7c^2 + 7c + 6\)
From the options, the polynomial \(-4c^2 + 4c + 6\) exactly matches the simplified form of the given polynomial.
Therefore, the quadratic function that is equivalent to the given polynomial is:
[tex]\[
\boxed{-4c^2 + 4c + 6}
\][/tex]
