Multiply the following. \[ -6 a^{2} b\left(2 b^{2}-5 b+19\right) \]
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To multiply the expression \(-6 a^{2} b(2 b^{2}-5 b+19)\), we'll distribute \(-6 a^{2} b\) to each term inside the parentheses. First, we multiply by \(2 b^{2}\): \[ -6 a^{2} b \cdot 2 b^{2} = -12 a^{2} b^{3} \] Next, we multiply by \(-5 b\): \[ -6 a^{2} b \cdot (-5 b) = 30 a^{2} b^{2} \] Finally, we multiply by \(19\): \[ -6 a^{2} b \cdot 19 = -114 a^{2} b \] Now, we combine all the products: \[ -12 a^{2} b^{3} + 30 a^{2} b^{2} - 114 a^{2} b \] Thus, the final result is: \[ -12 a^{2} b^{3} + 30 a^{2} b^{2} - 114 a^{2} b \]