act-dog

A. Incorrect. For any product of two binomials $\left(x+a\right)\left(x+b\right)=x^2+\left(a+b\right)x+ab$, the coefficient of the linear term will be $a+b$ and the constant term will be the product $ab$. You may have known $ab=-30$ needed to be true but forgot that $a+b=-1$ also needed to be true.

B. Incorrect. For any product of two binomials $\left(x+a\right)\left(x+b\right)=x^2+\left(a+b\right)x+ab$, the coefficient of the linear term will be $a+b$ and the constant term will be the product $ab$. You may have known $ab=-30$ needed to be true but thought that $a+b=1$ also needed to be true. Instead, $a+b=-1$ needs to be true.

C. Correct. Note that the trinomial $x^2-x-30$ can be factored as the product of two binomials. For any product of two binomials $\left(x+a\right)\left(x+b\right)=x^2+\left(a+b\right)x+ab$, the coefficient of the linear term will be $a+b$ and the constant term will be the product $ab$. To factor the quadratic polynomial $x^2-x-30$, identify two real numbers $a$ and $b$ such that $a+b=-1$ and $ab=-30$. Among the answer choices, $a=-6$ and $b=5$ give the desired results. Therefore, $x^2-x-30=\left(x+\left(-6\right)\right)\left(x+5\right)=\left(x-6\right)\left(x+5\right)$.

D. Incorrect. You may have thought that the factors of a trinomial would be the constant term divided by $2$, $\frac{30}{2}=15$. Instead, try identifying a product of two binomials, $\left(x+a\right)\left(x+b\right)$, which simplify to yield the given trinomial.