To find the product of the expressions [tex]\(2x + y\)[/tex] and [tex]\(5x - y + 3\)[/tex], let's follow the steps for polynomial multiplication.
Given:
[tex]\[
(2x + y) \cdot (5x - y + 3)
\][/tex]
We use the distributive property to expand this product.
1. Distribute [tex]\(2x\)[/tex] to each term inside the second parenthesis:
[tex]\[
2x \cdot (5x - y + 3) = 2x \cdot 5x + 2x \cdot (-y) + 2x \cdot 3
\][/tex]
Calculating each term:
[tex]\[
2x \cdot 5x = 10x^2
\][/tex]
[tex]\[
2x \cdot (-y) = -2xy
\][/tex]
[tex]\[
2x \cdot 3 = 6x
\][/tex]
So, the result of distributing [tex]\(2x\)[/tex] is:
[tex]\[
10x^2 - 2xy + 6x
\][/tex]
2. Distribute [tex]\(y\)[/tex] to each term inside the second parenthesis:
[tex]\[
y \cdot (5x - y + 3) = y \cdot 5x + y \cdot (-y) + y \cdot 3
\][/tex]
Calculating each term:
[tex]\[
y \cdot 5x = 5xy
\][/tex]
[tex]\[
y \cdot (-y) = -y^2
\][/tex]
[tex]\[
y \cdot 3 = 3y
\][/tex]
So, the result of distributing [tex]\(y\)[/tex] is:
[tex]\[
5xy - y^2 + 3y
\][/tex]
3. Now, we combine all the terms from steps 1 and 2:
[tex]\[
(10x^2 - 2xy + 6x) + (5xy - y^2 + 3y)
\][/tex]
4. Combine like terms:
[tex]\[
10x^2 + (-2xy + 5xy) + 6x - y^2 + 3y
\][/tex]
[tex]\[
10x^2 + 3xy + 6x - y^2 + 3y
\][/tex]
Therefore, the product of [tex]\(2x + y\)[/tex] and [tex]\(5x - y + 3\)[/tex] is:
[tex]\[
10x^2 + 3xy + 6x - y^2 + 3y
\][/tex]
So, filling in the blanks in the equation provided, we get:
[tex]\[
x^2 + \boxed{9} + \boxed{} + 3xy + \boxed{-1} y^2 + \boxed{3} y \boxed{10x^2} + x - \boxed{-2xy} \boxed{}
\][/tex]
