SAT Question efc469c4: Equivalent Expressions answer and explanation with Desmos solution

Choice A is correct. Expanding each squared binomial in the given expression yields $-14(25x^2-20x+4)+6(25x^2-30x+9)$. Applying the distributive property to both terms of this expression yields $(-350x^2+280x-56)+(150x^2-180x+54)$. Combining like terms in this expression yields $-200x^2+100x-2$. Since the given expression can be rewritten as $\frac{a}{6}{x}^2+\frac{b}{6}x+\frac{c}{6}$, it follows that $\frac{a}{6}=-200$, $\frac{b}{6}=100$, and $\frac{c}{6}=-2$. Multiplying both sides of each of these equations by $6$ yields $a=-1,200$, $b=600$, and $c=-12$, respectively. Substituting $-1,200$ for $a$, $600$ for $b$, and $-12$ for $c$ in the expression $a+b+c$ yields $-1,200+600-12$, or $-612$.