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: $\alpha \ast \beta =\frac{k}{1}$ 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: $1\ast 4=k$. Simplifying this, we get: $k=4$.