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Advanced Math / Equivalent expressions Difficulty: Hard

The expression $(3 x - 23) (19 x + 6)$ is equivalent to the expression $a x^{2} + b x + c$ , where $a$ , $b$ , and $c$ are constants. What is the value of $b$ ?

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Explanation

The correct answer is $negative 419$ . It's given that the expression $(3 x - 23) (19 x + 6)$ is equivalent to the expression $a x^{2} + b x + c$ , where $a$ , $b$ , and $c$ are constants. Applying the distributive property to the given expression, $(3 x - 23) (19 x + 6)$ , yields $(3 x) (19 x) + (3 x) (6) - (23) (19 x) - (23) (6)$ , which can be rewritten as $57 x^{2} + 18 x - 437 x - 138$ . Combining like terms yields $57 x^{2} - 419 x - 138$ . Since this expression is equivalent to $a x^{2} + b x + c$ , it follows that the value of $b$ is $negative 419$ .