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$(ax+3)(5x^{2}-bx+4)=20x^{3}-9x^{2}-2x+12$

The equation above is true for all $x$, where $a$ and $b$ are constants. What is the value of $ab$?

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$(ax+3)(5x^{2}-bx+4)=20x^{3}-9x^{2}-2x+12$

The equation above is true for all $x$, where $a$ and $b$ are constants. What is the value of $ab$?

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Choice C is correct. If the equation is true for all $x$, then the expressions on both sides of the equation will be equivalent. Multiplying the polynomials on the left-hand side of the equation gives $5ax^{3}-abx^{2}+4ax+15x^{2}-3bx+12$. On the right-hand side of the equation, the only $x^{2}$ term is $-9x^{2}$. Since the expressions on both sides of the equation are equivalent, it follows that $-abx^{2}+15x^{2}=-9x^{2}$, which can be rewritten as $(-ab+15)x^{2}=-9x^{2}$. Therefore, $-ab+15=-9$, which gives $ab=24$.

Choice A is incorrect. If $ab=18$, then the coefficient of $x^{2}$ on the left-hand side of the equation would be $-18+15=-3$, which doesn’t equal the coefficient of $x^{2}$, $-9$, on the right-hand side. Choice B is incorrect. If $ab=20$, then the coefficient of $x^{2}$ on the left-hand side of the equation would be $-20+15=-5$, which doesn’t equal the coefficient of $x^{2}$, $-9$, on the right-hand side. Choice D is incorrect. If $ab=40$, then the coefficient of $x^{2}$ on the left-hand side of the equation would be $-40+15=-25$, which doesn’t equal the coefficient of $x^{2}$, $-9$, on the right-hand side.

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$(ax+3)(5x^{2}-bx+4)=20x^{3}-9x^{2}-2x+12$

The equation above is true for all $x$, where $a$ and $b$ are constants. What is the value of $ab$?

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$(ax+3)(5x^{2}-bx+4)=20x^{3}-9x^{2}-2x+12$The equation above is true for all $x$, where $a$ and $b$ are constants. What is the value of $ab$?

$(ax+3)(5x^{2}-bx+4)=20x^{3}-9x^{2}-2x+12$The equation above is true for all $x$, where $a$ and $b$ are constants. What is the value of $ab$?

$(ax+3)(5x^{2}-bx+4)=20x^{3}-9x^{2}-2x+12$The equation above is true for all $x$, where $a$ and $b$ are constants. What is the value of $ab$?

$(ax+3)(5x^{2}-bx+4)=20x^{3}-9x^{2}-2x+12$

The equation above is true for all $x$, where $a$ and $b$ are constants. What is the value of $ab$?

$(ax+3)(5x^{2}-bx+4)=20x^{3}-9x^{2}-2x+12$

The equation above is true for all $x$, where $a$ and $b$ are constants. What is the value of $ab$?

$(ax+3)(5x^{2}-bx+4)=20x^{3}-9x^{2}-2x+12$

The equation above is true for all $x$, where $a$ and $b$ are constants. What is the value of $ab$?