Given that the expression \(ax^2 + bx + c\) equals 5 at x = 1, we can substitute x = 1 into the equation to get:

\(a(1)^2 + b(1) + c = 5\)

\(a + b + c = 5\)

The derivative of the expression \(ax^2 + bx + c\) is \(2ax + b\). Since the derivative is given as \(2x + 1\), we can equate the two expressions to get:

2ax + b = 2x + 1

Comparing the coefficients of \(x\) on both sides, we have \(2a = 2\), which gives us \(a = 1\).

Comparing the constant terms on both sides, we have b = 1. Substituting these values of a and b into the equation a + b + c = 5, we get:

1 + 1 + c = 5

c = 3

Therefore, the values of a, b, and c are respectively 1, 1, and 3.