Write an equivalent expression in standard form for
(6z-3)(8z+9)-13(45-7z-z2)

In the second step, none of those properties are being used. That is the distributive property. When you have a number being multiplied by parentheses, and inside the parentheses is a variable, you can apply the multiple to each term in the parentheses. None of the given properties apply there.

In 60 = -15x, the property being used there is the addition property of equality.

You are adding 6 to the right side, and so you're also adding it to the left side. Doing so does not skew the equation, and it still remains equal.

In the last part, -4 = x, the division property of equality is being used.

If we divide both sides by the same amount, the equation remains equal.

Dividing x by -15 isolates the variable and gives the solution.