1. **State the problem:** We want to find which expression is equivalent to the polynomial $$x^3 + 3x^2 + x + 3$$. 2. **Recall the distributive property and factoring:** To check equivalence, we can expand each given expression and compare it to the original polynomial. 3. **Check option A: $$(x^2 + 1)(x + 3)$$** Expand: $$\begin{aligned} (x^2 + 1)(x + 3) &= x^2 \cdot x + x^2 \cdot 3 + 1 \cdot x + 1 \cdot 3 \\ &= x^3 + 3x^2 + x + 3 \end{aligned}$$ This matches the original polynomial exactly. 4. **Check option B: $$(x^2 + 3)(x + 1)$$** Expand: $$\begin{aligned} (x^2 + 3)(x + 1) &= x^2 \cdot x + x^2 \cdot 1 + 3 \cdot x + 3 \cdot 1 \\ &= x^3 + x^2 + 3x + 3 \end{aligned}$$ This is $$x^3 + x^2 + 3x + 3$$, which is different from the original. 5. **Check option C: $$x(x + 1)(x + 3)$$** First expand $$(x + 1)(x + 3)$$: $$\begin{aligned} (x + 1)(x + 3) &= x^2 + 3x + x + 3 = x^2 + 4x + 3 \end{aligned}$$ Now multiply by $$x$$: $$\begin{aligned} x(x^2 + 4x + 3) &= x^3 + 4x^2 + 3x \end{aligned}$$ This is $$x^3 + 4x^2 + 3x$$, which is different from the original. 6. **Check option D: $$(x + 1)(x + 1)(x + 3)$$** First expand $$(x + 1)(x + 1)$$: $$\begin{aligned} (x + 1)^2 &= x^2 + 2x + 1 \end{aligned}$$ Now multiply by $$(x + 3)$$: $$\begin{aligned} (x^2 + 2x + 1)(x + 3) &= x^2 \cdot x + x^2 \cdot 3 + 2x \cdot x + 2x \cdot 3 + 1 \cdot x + 1 \cdot 3 \\ &= x^3 + 3x^2 + 2x^2 + 6x + x + 3 \\ &= x^3 + 5x^2 + 7x + 3 \end{aligned}$$ This is $$x^3 + 5x^2 + 7x + 3$$, which is different from the original. **Final answer:** Option A, $$(x^2 + 1)(x + 3)$$, is equivalent to $$x^3 + 3x^2 + x + 3$$.