Chapter 5.2, Problem 3MC

Consider the argument to show that $\left(-a\right)\left(b\right)=-\left(ab\right)$ for all integers $a$ and $b$:I know that $\left(-1\right)a=-a$ and $a$ could be variable to be replaced by $ab$ so that $\left(-1\right)ab=-\left(ab\right)$ and $\left(-1\right)ab=\left(-1\times a\right)b$ by associative property .Also $\left(-1.a\right)b=\left(-a\right)$ therefore $\left(-a\right)\left(b\right)=-\left(ab\right)$ If the argument is valid, give reasons for each step, if it is not valid explain for the same.