Answer
$F$
Work Step by Step
First of all, we see that we can factor $x$ from each of the terms in this expression:
$x(2x^2 + 5x - 12)$
Let's turn our attention to the quadratic polynomial within the parentheses.
To factor a quadratic polynomial in the form $ax^2 + bx + c$, we look at factors of $(a)(c)$ such that, when added together, equal the $b$ term.
For the expression $2x^2 + 5x - 12$, we look for the factors that, when multiplied together, will equal $(a)(c)$, which is $(2)(-12)$ or $-24$, but when adding the factors together will equal $b$ or $5$. We need one factor to be negative and the other to be positive, with the positive number to have a greater absolute value. When a negative number is multiplied with a positive number, the result is a negative number; however, when added together, they can make either a positive or a negative number, depending on the sign of the number with the greater absolute value. Here are some factors we came up with:
$(a)(c)$ = $(24)(-1)$
$b = 23$
$(a)(c)$ = $(12)(-2)$
$b = 10$
$(a)(c)$ = $(8)(-3)$
$b = 5$
The third pair works. We will use that pair to split the middle term:
$x(2x^2 + 8x - 3x - 12)$
Now, we can factor by grouping. We group the first two terms together and the second two terms together:
$x(2x^2 + 8x) + (-3x - 12)$
We see that $2x$ is a common factor for the first group, and $-3$ is a common factor for the second group, so let's factor those out:
$x[2x(x + 4) - 3(x + 4)]$
We see that $x + 4$ is common to both groups, so we put that binomial in parentheses. The other binomial will be $2x - 3$, which is composed of the coefficients in front of the binomials. We now have the two factors:
$x(x + 4)(2x - 3)$
This corresponds to option $F$.
