First, look at the coefficients, 6, 24, -72. 6 is a factor of 6, 24, and -72. Since one of the coefficients is in fact 6, then 6 is the GCF of the coefficinets.
Now look at the variables. You have x^4, x^3, and x^2. Remember what the exponents mean. They mean a multiplication of several factors of x.
x^4, x^3, x^2 mean xxxx, xxx, xx
xxxx, xxx, xx have 2 factors of x in common, so the greatest common factor of the variables is xx which is the same as x^2.
When you put together the common factor of the coefficients and the common factor of the variable parts, you get GCF = 6x^2.