You know the length of the rectangle is $\displaystyle 2x+3$ and the area of the rectangle is $\displaystyle 2x^2+5x+3$ that means the breadth of the rectangle has to look like $\displaystyle ax+b$, therefore:
$\displaystyle (2x+3)\cdot(ax+b)=2x^2+5x+3$
Determine $\displaystyle a$ and $\displaystyle b$
go learn how to do long division of polynomials ...
Polynomial Long Division
I never denied that. But the subject of the page is on polynomial long division, and I think that's asking the OP to run before they walk. With the UK system, there are about 3 years of teaching between this, which is applying the knowledge of factoring, and polynomial division.