To determine the number of turning points of the function \( m(x) = (2x+5)(x-3)^3(8x+1)^5 \), we need to analyze the polynomial.
The function is a product of several factors, and the degree of the polynomial can be found by adding the degrees of each factor:
- \( (2x + 5) \) is a linear factor of degree 1.
- \( (x - 3)^3 \) is a polynomial factor of degree 3.
- \( (8x + 1)^5 \) is a polynomial factor of degree 5.
Now, we calculate the total degree:
- The degree contributed by \( (2x + 5) \) is 1.
- The degree contributed by \( (x - 3)^3 \) is 3.
- The degree contributed by \( (8x + 1)^5 \) is 5.
Adding these together, we have: \[ 1 + 3 + 5 = 9 \]
For a polynomial of degree \( n \), the maximum number of turning points is \( n - 1 \). Therefore, for a polynomial of degree 9, the maximum number of turning points is: \[ 9 - 1 = 8 \]
Thus, Brandy should include eight at most turning points in her sketch.
So the correct response is: eight at most.