SAT Question a391ed22: Equivalent Expressions answer and explanation with Desmos solution

The expression $\left(\frac{1}{2}x+\frac{3}{2}\right)\left(\frac{3}{2}x+\frac{1}{2}\right)$ can be written in the form $a{x}^2+bx+c$, where $a$, $b$, and $c$ are constants, by multiplying out the expression using the distributive property of multiplication over addition. The result is $\frac{1}{2}x\times \frac{3}{2}x+\frac{1}{2}x\times \frac{1}{2}+\frac{3}{2}\times \frac{3}{2}x+\frac{3}{2}\times \frac{1}{2}$. This expression can be rewritten by multiplying as indicated to give $\frac{3}{4}{x}^2+\frac{1}{4}x+\frac{9}{4}x+\frac{3}{4}$, which can be simplified to $\frac{3}{4}{x}^2+\frac{10}{4}x+\frac{3}{4}$, or $\frac{3}{4}{x}^2+\frac{5}{2}x+\frac{3}{4}$. This is in the form $a{x}^2+bx+c$, where the value of $b$ is $\frac{5}{2}$. Note that 5/2, 2.5, and 2.50 are examples of ways to enter a correct answer.