27) $$ x^{2}+10 x y+16 y^{2} $$

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UpStudy AI · Accepted Answer

1. Recognize that the quadratic can be written in the form
   $$
   (x + py)(x + qy)
   $$
   where expanding gives
   $$
   x^2 + (p+q)xy + pq\,y^2.
   $$
   
2. Compare with the given expression
   $$
   x^2 + 10xy + 16y^2.
   $$
   We have the system:
   $$
   p+q=10 \quad \text{and} \quad pq=16.
   $$

3. Find numbers $$ p $$ and $$ q $$ that satisfy:
   $$
   p+q=10 \quad \text{and} \quad pq=16.
   $$
   The numbers $$ 2 $$ and $$ 8 $$ work since:
   $$
   2+8=10 \quad \text{and} \quad 2\cdot8=16.
   $$

4. Therefore, the factorization is:
   $$
   (x+2y)(x+8y).
   $$
The expression $$ x^{2} + 10xy + 16y^{2} $$ can be factored as $$ (x + 2y)(x + 8y) $$.

27) \( x^{2}+10 x y+16 y^{2} \)

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