Sure, let's simplify the expression step-by-step:
Expression:
[tex]\[
\left(\frac{3}{2}\right)^2 + 4 \div 2 \cdot 3
\][/tex]
1. Simplify the expression within the parentheses:
[tex]\[
\left(\frac{3}{2}\right)^2 = \frac{3}{2} \times \frac{3}{2} = \frac{9}{4}
\][/tex]
2. Handle the division and multiplication according to the order of operations (PEMDAS/BODMAS):
[tex]\[
4 \div 2 \cdot 3
\][/tex]
First do the division:
[tex]\[
4 \div 2 = 2
\][/tex]
Then, multiply the result by 3:
[tex]\[
2 \cdot 3 = 6
\][/tex]
3. Add the results of the two parts:
[tex]\[
\frac{9}{4} + 6
\][/tex]
To add these, we need to convert 6 into a fraction with a common denominator:
[tex]\[
6 = \frac{24}{4}
\][/tex]
So the addition becomes:
[tex]\[
\frac{9}{4} + \frac{24}{4} = \frac{33}{4}
\][/tex]
Final answer:
[tex]\(\boxed{\frac{33}{4}}\)[/tex]
