Let's tackle the problem step by step.
We want to multiply [tex]\((x + 3)\)[/tex] by [tex]\((2x + 5)\)[/tex] and find the coefficient of [tex]\(x\)[/tex] in the resulting expression.
1. Distribute both terms in the first binomial to each term in the second binomial. This process is known as the FOIL method (First, Outside, Inside, Last):
[tex]\[
(x + 3)(2x + 5)
\][/tex]
2. Multiply the terms:
- First: Multiply the first terms in each binomial:
[tex]\[
x \times 2x = 2x^2
\][/tex]
- Outside: Multiply the outer terms:
[tex]\[
x \times 5 = 5x
\][/tex]
- Inside: Multiply the inner terms:
[tex]\[
3 \times 2x = 6x
\][/tex]
- Last: Multiply the last terms in each binomial:
[tex]\[
3 \times 5 = 15
\][/tex]
3. Combine all these products together:
[tex]\[
2x^2 + 5x + 6x + 15
\][/tex]
4. Combine like terms (specifically the [tex]\(x\)[/tex] terms):
[tex]\[
5x + 6x = 11x
\][/tex]
So the expression simplifies to:
[tex]\[
2x^2 + 11x + 15
\][/tex]
The coefficient of the [tex]\(x\)[/tex]-term in this expression is [tex]\(11\)[/tex].
Thus, the coefficient of [tex]\(x\)[/tex] when [tex]\((x + 3)\)[/tex] is multiplied by [tex]\((2x + 5)\)[/tex] is:
[tex]\[
\boxed{11}
\][/tex]
