To solve the given division problem involving fractions, we will follow a step-by-step approach to simplify the expression:
[tex]\[
\frac{3}{7 x^2 - q} \div \frac{12}{21 x^2 - 3 q}
\][/tex]
### Step 1: Rewrite the Division as Multiplication by the Reciprocal
Recall that dividing by a fraction is the same as multiplying by its reciprocal. Hence, we rewrite the division as follows:
[tex]\[
\frac{3}{7 x^2 - q} \div \frac{12}{21 x^2 - 3 q} = \frac{3}{7 x^2 - q} \times \frac{21 x^2 - 3 q}{12}
\][/tex]
### Step 2: Multiply the Numerators and Denominators
Now, we multiply the numerators together and the denominators together:
[tex]\[
\frac{3 \times (21 x^2 - 3 q)}{(7 x^2 - q) \times 12}
\][/tex]
### Step 3: Simplify the Expression
First, distribute the 3 in the numerator:
[tex]\[
3 \times (21 x^2 - 3 q) = 63 x^2 - 9 q
\][/tex]
Thus, the expression becomes:
[tex]\[
\frac{63 x^2 - 9 q}{12 (7 x^2 - q)}
\][/tex]
### Step 4: Factor Common Terms if Possible
Let's factor out the common factors in both the numerator and the denominator:
In the numerator, [tex]\(63 x^2 - 9 q\)[/tex]:
[tex]\[
63 x^2 - 9 q = 9 \times (7 x^2 - q)
\][/tex]
In the denominator, [tex]\(12 (7 x^2 - q)\)[/tex]:
Clearly, it is already factored as [tex]\(12 \times (7 x^2 - q)\)[/tex].
So, the simplified expression now looks like:
[tex]\[
\frac{9 \times (7 x^2 - q)}{12 \times (7 x^2 - q)}
\][/tex]
### Step 5: Cancel Out the Common Factor
We see that the term [tex]\((7 x^2 - q)\)[/tex] is present in both the numerator and the denominator, so they cancel each other out:
[tex]\[
\frac{9 \times \cancel{(7 x^2 - q)}}{12 \times \cancel{(7 x^2 - q)}} = \frac{9}{12}
\][/tex]
### Step 6: Simplify the Fraction
Finally, simplify [tex]\(\frac{9}{12}\)[/tex]:
[tex]\[
\frac{9}{12} = \frac{3}{4}
\][/tex]
### Conclusion
The simplified result of the given problem is:
[tex]\[
\frac{3}{4}
\][/tex]
