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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.

Solve the quadratic equation x^2 + 2(m-1)x + (m+5) = 0

Learn how to solve the quadratic equation x^2 + 2(m-1)x + (m+5) using the quadratic formula. Explore the impact of the parameter m on the solutions and apply algebraic concepts to find the roots.

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