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The problem given is:

The polynomial \( 5x^m + (k-1)x^2 - (n+4)x - 3 \) is a cubic polynomial in \( x \), with the coefficient of the quadratic term equal to 1. You are asked to find the value of \( m - k + n \).

### Step-by-step solution:

1. **Determine the degree of the polynomial:**
   - The polynomial is given as \( 5x^m + (k-1)x^2 - (n+4)x - 3 \). For it to be a cubic polynomial, the highest power of \( x \) must be 3. Hence, \( m = 3 \).

2. **Use the condition for the quadratic term:**
   - The coefficient of the \( x^2 \) term is \( (k-1) \). It is given that the quadratic coefficient must be 1. Therefore, we set:
     \[
     k - 1 = 1
     \]
     Solving for \( k \):
     \[
     k = 2
     \]

3. **Final step – find \( m - k + n \):**
   - Now, substitute \( m = 3 \) and \( k = 2 \) into the expression for \( m - k + n \):
     \[
     m - k + n = 3 - 2 + n = 1 + n
     \]

   Thus, \( m - k + n = 1 + n \).

### Answer:
The value of \( m - k + n \) is \( 1 + n \).

Would you like more details on any part of the solution?

Here are 5 related questions to deepen your understanding:
1. What is a cubic polynomial, and why must \( m = 3 \)?
2. How do we determine the coefficient of the quadratic term in a polynomial?
3. Can you solve for \( m \), \( k \), and \( n \) if additional conditions are given?
4. What happens to the polynomial if the cubic term is absent (i.e., \( m < 3 \))?
5. How would the problem change if the constant term were also dependent on variables?

**Tip:** When solving polynomial equations, always focus on matching the degrees and coefficients with the given conditions for quicker resolution.

The polynomial 5x^m + (k-1)x^2 - (n+4)x - 3 is a cubic polynomial in x, with the coefficient of the quadratic term equal to 1. You are asked to find the value of m - k + n.

Solve the cubic polynomial equation 5x^m + (k-1)x^2 - (n+4)x - 3 by determining the values of m, k, and n. Learn about polynomial degrees, coefficient comparison, and solve for m - k + n. Suitable for Grades 10-12.

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