Certainly! Let's work through the division step-by-step to simplify the given expression:
[tex]\[
\frac{6 t^3 + 15 t^2}{-3 t}
\][/tex]
### Step 1: Simplify the Expression
First, recognize that both the numerator and the denominator can be divided by the common factor [tex]\(-3 t\)[/tex]. To do this, we will break down the terms in the numerator individually divided by the denominator:
1. Divide [tex]\(6 t^3\)[/tex] by [tex]\(-3 t\)[/tex]:
[tex]\[
\frac{6 t^3}{-3 t} = \frac{6}{-3} \cdot \frac{t^3}{t} = -2 t^2
\][/tex]
2. Divide [tex]\(15 t^2\)[/tex] by [tex]\(-3 t\)[/tex]:
[tex]\[
\frac{15 t^2}{-3 t} = \frac{15}{-3} \cdot \frac{t^2}{t} = -5 t
\][/tex]
### Step 2: Combine the Results
Now, we will combine the simplified results:
[tex]\[
-2 t^2 - 5 t
\][/tex]
### Step 3: Factor the Result
Observe the expression [tex]\(-2 t^2 - 5 t\)[/tex], which has a common factor of [tex]\(t\)[/tex]. Thus, we can factor out [tex]\(t\)[/tex] from each term:
[tex]\[
t(-2 t - 5)
\][/tex]
### Final Answer
Hence, the simplified form of the given expression is:
[tex]\[
\frac{6 t^3 + 15 t^2}{-3 t} = t(-2 t - 5)
\][/tex]
