To divide the given rational expressions:
[tex]\[
\frac{4x + 14}{x^6} \div \frac{6x^3 + 21x^2}{x^3}
\][/tex]
Here are the steps:
### Step 1: Write the division as multiplication by the reciprocal.
When we divide by a fraction, it is equivalent to multiplying by its reciprocal. Therefore, we convert the division problem into a multiplication problem:
[tex]\[
\frac{4x + 14}{x^6} \div \frac{6x^3 + 21x^2}{x^3} = \frac{4x + 14}{x^6} \times \frac{x^3}{6x^3 + 21x^2}
\][/tex]
### Step 2: Simplify the multiplication of the fractions.
To perform the multiplication, we multiply the numerators together and the denominators together:
[tex]\[
\frac{4x + 14}{x^6} \times \frac{x^3}{6x^3 + 21x^2} = \frac{(4x + 14) \cdot x^3}{x^6 \cdot (6x^3 + 21x^2)}
\][/tex]
### Step 3: Combine and simplify the expressions.
Now, we combine the terms in the numerator and denominator:
[tex]\[
\text{Numerator: } (4x + 14) \cdot x^3 = 4x^4 + 14x^3
\][/tex]
[tex]\[
\text{Denominator: } x^6 \cdot (6x^3 + 21x^2) = 6x^9 + 21x^8
\][/tex]
Thus, the simplified rational expressions are:
[tex]\[
\boxed{4x^4 + 14x^3} \quad\text{and}\quad \boxed{6x^9 + 21x^8}
\][/tex]
However, for further simplification in terms of factoring common factors, we have:
[tex]\[
4x^4 + 14x^3 = x^3(4x + 14)\quad \text{and} \quad 6x^9 + 21x^8 = x^8(6x + 21)
\][/tex]
### Final Result:
After simplification, the numerator and the denominator of the resulting expression are:
[tex]\[
\boxed{x^3(4x + 14)} \quad \text{and} \quad \boxed{x^8(6x + 21)}
\][/tex]
