The given equation is:

$x^2 + 2(m - 1)x + (m + 5) = 0$

This is a quadratic equation in $x$, where $m$ is a parameter. To solve this equation, we can apply the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the equation $ax^2 + bx + c = 0$, we identify the coefficients as:

• $a = 1$,
• $b = 2(m - 1)$,
• $c = m + 5$.

Substituting into the quadratic formula:

$x = \frac{-2(m - 1) \pm \sqrt{[2(m - 1)]^2 - 4(1)(m + 5)}}{2(1)}$

Let me simplify this for you.The solutions to the quadratic equation $x^2 + 2(m - 1)x + (m + 5) = 0$ are:

$x_1 = -m - \sqrt{(m - 4)(m + 1)} + 1$ $x_2 = -m + \sqrt{(m - 4)(m + 1)} + 1$

These are the two roots of the equation in terms of the parameter $m$.

Would you like further details on this solution, or do you have any other questions?

Here are 5 related questions for you:

1. How can the discriminant help determine the nature of the roots?
2. What happens to the solutions when $m = 4$?
3. Can we simplify the solutions further for any specific values of $m$?
4. How does the quadratic formula change for different values of the coefficient $a$?
5. What are the conditions for having real, equal, or complex roots in a quadratic equation?

Tip: Always check the discriminant $b^2 - 4ac$ when analyzing quadratic equations, as it reveals important information about the nature of the roots.