Answer
Due to the Order of Operations, work inside parentheses is done before exponents are used. Because $-1\times5$ in the first example is inside the parentheses, you must multiply before you use exponents. This results in $-5$ to the fourth power, which gives an answer of $625$. In the second example given no parentheses are present, which means we must solve exponents before we multiple by the $-1$. This results in us solving $-1\times5^{4}$, which equals $-625$.
Work Step by Step
$(-1\times5)^4$ Multiply $5$ by $-1$
$(-5)^4$ Take $-5$ to the fourth power
$(-5\times-5\times-5\times-5)$ Multiply $-5$ by itself four times
$(25\times25)$ Multiple $25$ by $25$
$625$
$-1\times5^4$ Take $5$ to the fourth power
$-1\times(5\times5\times5\times5)$ Multiply $5$ by itself four times
$-1\times(25\times25)$ Multiple $25$ by $25$
$-1\times625$ Multiply by $-1$
$-625$
