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The term $a*b$ is nonnegative whenever both $a$ and $b$ have the same sign. It is negative whenever $a$ is positive and $b$ is negative.   Since $a$, $b$ are integers with $a = b+10$, these are all the possible ways that $ab$ can be negative:
\begin{eqnarray*}
    a = 1, b = -9, ab = -9 \\
    a = 2, b = -8, ab = -16 \\
    a = 3, b = -7, ab = -21 \\
    a = 4, b = -6, ab = -24 \\
    a = 5, b = -5, ab = -25 \\
    a = 6, b = -4, ab = -24 \\
    a = 7, b = -3, ab = -21 \\
    a = 8, b = -2, ab = -16 \\
    a = 9, b = -1, ab = -9
\end{eqnarray*}

So the minimum value of $ab$ is $-25$.