To rewrite the polynomial [tex]\( 9x^2 + 3xy + 15x + 5y \)[/tex] in factored form, follow these steps:
1. Identify and group terms: We begin by examining the given polynomial [tex]\( 9x^2 + 3xy + 15x + 5y \)[/tex].
2. Factor by grouping: To factor by grouping, we look for common factors in parts of the polynomial.
[tex]\( 9x^2 + 3xy + 15x + 5y \)[/tex]
Notice that we can group the terms as [tex]\((9x^2 + 3xy)\)[/tex] and [tex]\((15x + 5y)\)[/tex]:
[tex]\[ 9x^2 + 3xy + 15x + 5y = 3x(3x + y) + 5(3x + y) \][/tex]
Here, [tex]\(3x\)[/tex] is a common factor in the first group, and [tex]\(5\)[/tex] is a common factor in the second group.
3. Factor out the common binomial: Now, notice that [tex]\((3x + y)\)[/tex] is common in both groups:
[tex]\[ 3x(3x + y) + 5(3x + y) = (3x + y)(3x + 5) \][/tex]
The polynomial can now be written in factored form as [tex]\((3x + y)(3x + 5)\)[/tex].
Therefore, the correct answer is:
[tex]\[
(3x + 5)(3x + y)
\][/tex]
