Certainly! Let's perform the multiplication of the expressions \( (5x^2 + 7xy + 10y^2) \times (5x - 7y) \) step by step using the vertical method.
First, we'll write out the expressions to be multiplied:
[tex]\[
\begin{array}{c}
(5 x^2 + 7 x y + 10 y^2) \\
\times (5 x - 7 y) \\
\end{array}
\][/tex]
Now, we will decompose and multiply each term in the polynomial \( 5x - 7y \) by each term in the polynomial \( 5x^2 + 7xy + 10y^2 \).
1. Multiply each term in \( 5x^2 + 7xy + 10y^2 \) by \( 5x \):
[tex]\[
\begin{array}{c}
5x \cdot (5x^2 + 7xy + 10y^2) \\
= 5x \cdot 5x^2 + 5x \cdot 7xy + 5x \cdot 10y^2 \\
= 25x^3 + 35x^2y + 50xy^2
\end{array}
\][/tex]
2. Multiply each term in \( 5x^2 + 7xy + 10y^2 \) by \(-7y\):
[tex]\[
\begin{array}{c}
-7y \cdot (5x^2 + 7xy + 10y^2) \\
= -7y \cdot 5x^2 + -7y \cdot 7xy + -7y \cdot 10y^2 \\
= -35x^2y - 49xy^2 - 70y^3
\end{array}
\][/tex]
3. Combine the results:
We add the products obtained in the previous steps:
[tex]\[
(25x^3 + 35x^2y + 50xy^2) + (-35x^2y - 49xy^2 - 70y^3)
\][/tex]
Now, let's combine like terms to get the final expression:
[tex]\[
\begin{aligned}
&25x^3 + 35x^2y - 35x^2y + 50xy^2 - 49xy^2 - 70y^3 \\
=& 25x^3 + (35x^2y - 35x^2y) + (50xy^2 - 49xy^2) - 70y^3 \\
=& 25x^3 + 0x^2y + 1xy^2 - 70y^3 \\
=& 25x^3 + xy^2 - 70y^3
\end{aligned}
\][/tex]
Thus, the product of the given expressions is:
[tex]\[
(5x^2 + 7xy + 10y^2) \times (5x - 7y) = 25x^3 + xy^2 - 70y^3
\][/tex]
This completes our detailed, step-by-step solution.
