Sure, let's solve the expression [tex]\((2x + 3)(3x + 5)\)[/tex] step-by-step to expand it into a trinomial.
### Step-by-Step Solution
1. Distribute each term in the first binomial to every term in the second binomial:
When dealing with binomials, we use the distributive property (also known as the FOIL method for binomials which stands for First, Outer, Inner, Last).
[tex]\[
(2x + 3)(3x + 5)
\][/tex]
2. Multiply the terms:
- First: Multiply the first terms in each binomial:
[tex]\[
2x \cdot 3x = 6x^2
\][/tex]
- Outer: Multiply the outer terms in the binomial:
[tex]\[
2x \cdot 5 = 10x
\][/tex]
- Inner: Multiply the inner terms in the binomial:
[tex]\[
3 \cdot 3x = 9x
\][/tex]
- Last: Multiply the last terms in each binomial:
[tex]\[
3 \cdot 5 = 15
\][/tex]
3. Add all these products together:
[tex]\[
6x^2 + 10x + 9x + 15
\][/tex]
4. Combine like terms:
- Combine the [tex]\(x\)[/tex]-terms:
[tex]\[
10x + 9x = 19x
\][/tex]
- Thus, after combining these terms, you get:
[tex]\[
6x^2 + 19x + 15
\][/tex]
### Final Trinomial
So, the expanded form of the expression [tex]\((2x + 3)(3x + 5)\)[/tex] is:
[tex]\[
6x^2 + 19x + 15
\][/tex]
This trinomial is the product of multiplying the given binomials.
