Certainly! Let's simplify the given expression step by step.
We have the expression:
[tex]$
\frac{6x^3 + 3x^2 - 12x}{3x}
$[/tex]
### Step 1: Factor the numerator if possible
First, we'll look at the numerator [tex]\(6x^3 + 3x^2 - 12x\)[/tex] to see if we can factor out any common terms.
Notice that each term in the numerator has a factor of [tex]\(3x\)[/tex]:
[tex]\[ 6x^3 = 3x \cdot 2x^2 \][/tex]
[tex]\[ 3x^2 = 3x \cdot x \][/tex]
[tex]\[ -12x = 3x \cdot -4 \][/tex]
So we can factor [tex]\(3x\)[/tex] out of the entire numerator:
[tex]\[ 6x^3 + 3x^2 - 12x = 3x(2x^2 + x - 4) \][/tex]
### Step 2: Simplify the fraction
Now, substitute the factored form back into the original expression:
[tex]$
\frac{3x(2x^2 + x - 4)}{3x}
$[/tex]
Since [tex]\(3x\)[/tex] is a common factor in both the numerator and the denominator, we can cancel them out:
[tex]\[ = \frac{\cancel{3x}(2x^2 + x - 4)}{\cancel{3x}} \][/tex]
[tex]\[ = 2x^2 + x - 4 \][/tex]
### Conclusion:
After simplifying, we obtain:
[tex]\[ \boxed{2x^2 + x - 4} \][/tex]
So, the correct simplified form of the given expression is:
[tex]\[ 2x^2 + x - 4 \][/tex]
