Step 1 :The given line is parallel to the line $y=4x$. This means that the slope of the line we are looking for is also 4.
Step 2 :We are also given that the line contains the point $(-4,3)$. We can substitute this point into the equation $y=mx+b$ to find the y-intercept $b$.
Step 3 :Substituting the point $(-4,3)$ into the equation $y=mx+b$ gives us $3=4*(-4)+b$.
Step 4 :Solving this equation for $b$ gives us $b=19$.
Step 5 :Therefore, the equation of the line in slope-intercept form is $y=4x+19$.
Step 6 :\(\boxed{y=4x+19}\) is the final answer.