Given the quadratic equation \(x^2 + 5x + k = 0\) where \(k\) is the constant of variation. The two roots of the equation are 1 and 4. Find the value of \(k\).

Step 1 ：Step 1: We know that the sum of the roots of a quadratic equation is equal to the negative ratio of the coefficient of \(x\) to the coefficient of \(x^2\). So, from the given equation, we can write: \(\alpha + \beta = -\frac{b}{a}\) where \(\alpha\) and \(\beta\) are the roots of the equation, and \(a\) and \(b\) are the coefficients of \(x^2\) and \(x\) respectively.

Step 2 ：Step 2: Substituting the given values, we get: \(1 + 4 = -\frac{5}{1}\). Simplifying this, we get: \(5 = -5\). This is a contradiction, and hence, there is a mistake in our assumption.

Step 3 ：Step 3: The mistake in our assumption is that the roots of the equation are not 1 and 4. So, we need to find the correct roots of the equation.

Step 4 ：Step 4: The product of the roots of a quadratic equation is equal to the constant term divided by the coefficient of \(x^2\). So, from the given equation, we can write: \(\alpha * \beta = \frac{k}{1}\) where \(\alpha\) and \(\beta\) are the roots of the equation, and \(k\) is the constant term.

Step 5 ：Step 5: Substituting the given values, we get: \(1 * 4 = k\). Simplifying this, we get: \(k = 4\).