(e) $$ 6 s^{2}-7 s t-5 t^{2} $$

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UpStudy AI · Accepted Answer

1. We start with the expression
   $$
   6s^2 - 7st - 5t^2
   $$
2. Think of the expression as a quadratic in $$ s $$, with coefficients $$ a = 6 $$, $$ b = -7t $$, and $$ c = -5t^2 $$.

3. We need two numbers that multiply to
   $$
   a \cdot c = 6 \cdot (-5t^2) = -30t^2
   $$
   and add to
   $$
   -7t.
   $$
   Notice that
   $$
   3t + (-10t) = -7t \quad \text{and} \quad 3t \cdot (-10t) = -30t^2.
   $$

4. Rewrite the middle term $$ -7st $$ using $$ 3t $$ and $$-10t $$:
   $$
   6s^2 + 3st - 10st - 5t^2.
   $$

5. Group the terms:
   $$
   \left(6s^2 + 3st\right) - \left(10st + 5t^2\right).
   $$

6. Factor out the common factors in each group:
   $$
   3s(2s + t) - 5t(2s + t).
   $$

7. Notice that $$(2s + t)$$ is a common factor:
   $$
   (2s + t)(3s - 5t).
   $$

Thus, the factorized form is
$$
(2s + t)(3s - 5t).
$$
The expression $$6s^2 - 7st - 5t^2$$ can be factored as $$(2s + t)(3s - 5t)$$.

(e) \( 6 s^{2}-7 s t-5 t^{2} \)

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