## Expanding the Expression: (d^2 + 3)(d^2 + 2d + 1)

This problem involves expanding a product of two quadratic expressions. We can achieve this by using the distributive property (also known as FOIL).

**FOIL stands for:**

**F**irst: Multiply the first terms of each binomial.**O**uter: Multiply the outer terms of the binomials.**I**nner: Multiply the inner terms of the binomials.**L**ast: Multiply the last terms of each binomial.

**Let's apply FOIL to our expression:**

**(d^2 + 3)(d^2 + 2d + 1)**

**First:**(d^2)(d^2) =**d^4****Outer:**(d^2)(2d) =**2d^3****Inner:**(3)(d^2) =**3d^2****Last:**(3)(2d) =**6d****Last:**(3)(1) =**3**

**Now, combine the terms:**

**d^4 + 2d^3 + 3d^2 + 6d + 3**

**Therefore, the expanded form of (d^2 + 3)(d^2 + 2d + 1) is d^4 + 2d^3 + 3d^2 + 6d + 3.**