SAT Question 04887c7b: Equivalent Expressions answer and explanation with Desmos solution

Choice B is correct. It’s given that $f(x)=2x+3$, $g(x)=7x-2$, and $h(x)=5x+6$. It’s also given that $f\left(x\right)\cdot g\left(x\right)-h\left(x\right)=a{x}^2+bx+c$, where $a$, $b$, and $c$ are constants. Substituting $2x+3$ for $f(x)$, $7x-2$ for $g(x)$, and $5x+6$ for $h(x)$ in this equation yields $(2x+3)(7x-2)-(5x+6)=a{x}^2+bx+c$. Applying the distributive property to the left-hand side of this equation yields $(2x)(7x)+(2x)(-2)+(3)(7x)+(3)(-2)+(-1)(5x)+(-1)(6)=a{x}^2+bx+c$, or $14x^2-4x+21x-6-5x-6=a{x}^2+bx+c$. Combining like terms on the left-hand side of this equation yields $14x^2+12x-12=a{x}^2+bx+c$. It follows that the value of $b$ is $12$.