Identify the given roots and the general form of the quadratic equation.

The given quadratic equation is $x^2 + bx + c = 0$.
The roots are $x_1 = \frac{-7+\sqrt{61}}{2}$ and $x_2 = \frac{-7-\sqrt{61}}{2}$.
For a quadratic equation $ax^2 + bx + c = 0$, the sum of the roots is $x_1 + x_2 = -\frac{b}{a}$ and the product of the roots is $x_1 \cdot x_2 = \frac{c}{a}$.
In this case, $a=1$.

Since $x_1 + x_2 = -\frac{b}{a}$ and $a=1$:

This is in the form $(A+B)(A-B) = A^2 - B^2$, where $A = -7$ and $B = \sqrt{61}$.

Since $x_1 \cdot x_2 = \frac{c}{a}$ and $a=1$:

The values are $b=7$ and $c=-3$.

The final answer is $b=7, c=-3$.