To expand and simplify the expression [tex]\((3x + 4)(2x + 3)\)[/tex], we need to use the distributive property (also known as the FOIL method for binomials). This involves multiplying each term in the first binomial by each term in the second binomial and then combining like terms. Let's go through it step-by-step.
1. First terms: Multiply the first term in each binomial.
[tex]\[
3x \cdot 2x = 6x^2
\][/tex]
2. Outer terms: Multiply the first term in the first binomial by the second term in the second binomial.
[tex]\[
3x \cdot 3 = 9x
\][/tex]
3. Inner terms: Multiply the second term in the first binomial by the first term in the second binomial.
[tex]\[
4 \cdot 2x = 8x
\][/tex]
4. Last terms: Multiply the second term in each binomial.
[tex]\[
4 \cdot 3 = 12
\][/tex]
Now, we combine all these results:
[tex]\[
6x^2 + 9x + 8x + 12
\][/tex]
Next, we combine the like terms ([tex]\(9x\)[/tex] and [tex]\(8x\)[/tex]):
[tex]\[
6x^2 + 17x + 12
\][/tex]
Thus, the expanded and simplified form of [tex]\((3x + 4)(2x + 3)\)[/tex] is:
[tex]\[
6x^2 + 17x + 12
\][/tex]
This is the final answer.
