1. **State the problem:** We need to simplify and understand the function $y = (x + 1)(7\sqrt{x} + 2)$.\n\n2. **Recall the formula and rules:** The function is a product of two expressions: a linear term $(x+1)$ and a term involving a square root $7\sqrt{x} + 2$. Note that $\sqrt{x}$ is defined for $x \geq 0$.\n\n3. **Expand the expression:** Use the distributive property (FOIL) to multiply:\n$$y = (x + 1)(7\sqrt{x} + 2) = x \cdot 7\sqrt{x} + x \cdot 2 + 1 \cdot 7\sqrt{x} + 1 \cdot 2$$\n\n4. **Simplify each term:**\n- $x \cdot 7\sqrt{x} = 7x\sqrt{x} = 7x^{1}x^{\frac{1}{2}} = 7x^{\frac{3}{2}}$\n- $x \cdot 2 = 2x$\n- $1 \cdot 7\sqrt{x} = 7\sqrt{x} = 7x^{\frac{1}{2}}$\n- $1 \cdot 2 = 2$\n\n5. **Write the simplified function:**\n$$y = 7x^{\frac{3}{2}} + 2x + 7x^{\frac{1}{2}} + 2$$\n\n6. **Domain:** Since $\sqrt{x}$ is defined only for $x \geq 0$, the domain of $y$ is $[0, \infty)$.\n\n7. **Graph behavior:** The function increases as $x$ increases because the highest power term $7x^{\frac{3}{2}}$ dominates for large $x$. The graph starts at $y = (0+1)(7\sqrt{0}+2) = 1 \times 2 = 2$ when $x=0$.\n\n**Final answer:**\n$$y = 7x^{\frac{3}{2}} + 2x + 7x^{\frac{1}{2}} + 2, \quad x \geq 0$$