25) $$ m^{2}-15 m n+50 n^{2} $$

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UpStudy AI · Accepted Answer

1. We start with the expression:
   $$
   m^2 - 15mn + 50n^2
   $$

2. We look for two numbers that multiply to $$50$$ (the coefficient of $$n^2$$) and add up to $$-15$$ (the coefficient of $$mn$$). These numbers are $$-5$$ and $$-10$$, since:
   $$
   (-5) \times (-10) = 50 \quad \text{and} \quad (-5) + (-10) = -15.
   $$

3. We rewrite the middle term $$-15mn$$ using $$-5mn$$ and $$-10mn$$:
   $$
   m^2 - 5mn - 10mn + 50n^2.
   $$

4. We group the terms:
   $$
   (m^2 - 5mn) + (-10mn + 50n^2).
   $$

5. Factor out common factors in each group:
   $$
   m(m - 5n) - 10n(m - 5n).
   $$

6. Notice that $$(m - 5n)$$ is a common factor:
   $$
   (m - 5n)(m - 10n).
   $$

Thus, the factored form is:
$$
(m - 5n)(m - 10n).
$$
The expression $$ m^{2} - 15mn + 50n^{2} $$ can be factored as $$ (m - 5n)(m - 10n) $$.

25) \( m^{2}-15 m n+50 n^{2} \)

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