To divide the given rational expressions, we need to multiply the first rational expression by the reciprocal of the second rational expression. Specifically, we will multiply:
[tex]\[
\frac{6 s^2 + 7 s t - 5 t^2}{6 s^2 - 5 s t + t^2}
\][/tex]
by
[tex]\[
\frac{6 s^2 + 19 s t - 7 t^2}{3 s^2 + 26 s t + 35 t^2}
\][/tex]
So, the expression to solve becomes:
[tex]\[
\frac{6 s^2 + 7 s t - 5 t^2}{6 s^2 - 5 s t + t^2} \times \frac{6 s^2 + 19 s t - 7 t^2}{3 s^2 + 26 s t + 35 t^2}
\][/tex]
1. Factor each polynomial in the numerators and denominators if possible.
Let's examine the first numerator \(6 s^2 + 7 s t - 5 t^2\):
- By factoring, we find:
[tex]\[
6 s^2 + 7 s t - 5 t^2 = (3s - t)(2s + 5t)
\][/tex]
Next, the first denominator \(6 s^2 - 5 s t + t^2\):
- By factoring, we find:
[tex]\[
6 s^2 - 5 s t + t^2 = (3s - t)(2s - t)
\][/tex]
Now for the second numerator \(6 s^2 + 19 s t - 7 t^2\):
- By factoring, we find:
[tex]\[
6 s^2 + 19 s t - 7 t^2 = (3s + 7t)(2s - t)
\][/tex]
And finally, the second denominator \(3 s^2 + 26 s t + 35 t^2\):
- By factoring, we find:
[tex]\[
3 s^2 + 26 s t + 35 t^2 = (3s + 5t)(s + 7t)
\][/tex]
2. Rewrite the expression using the factored forms:
[tex]\[
\frac{(3s - t)(2s + 5t)}{(3s - t)(2s - t)} \times \frac{(3s + 7t)(2s - t)}{(3s + 5t)(s + 7t)}
\][/tex]
3. Cancel any common factors in the numerators and denominators:
- We see that \((3s - t)\) appears in both the numerator and denominator of the first fraction.
- \((2s - t)\) appears in both the numerator and denominator of the second fraction.
- \((3s + 5t)\) also appears both as a numerator and a denominator.
After canceling these common factors, we are left with:
[tex]\[
\frac{(2s + 5t)}{1} \times \frac{(3s + 7t)}{(s + 7t)}
\][/tex]
4. Simplify the remaining expression:
[tex]\[
\frac{(2s + 5t)(3s + 7t)}{(s + 7t)}
\][/tex]
Since there are no further common factors to cancel, the final result is:
[tex]\[
\frac{(2s + 5t)(3s + 7t)}{s + 7t} = 2s + 5t
\][/tex]
Thus, the solution to the given problem is:
[tex]\[
\boxed{\frac{(2s + 5t)(3s + 7t)}{s + 7t}}
\][/tex]
