1. **Problem Statement:** Expand each expression by distributing the term outside the parentheses to each term inside the parentheses. 2. **Formula Used:** For any expressions of the form $a(b + c)$, use the distributive property: $$a(b + c) = ab + ac$$ 3. **Step-by-step Expansion:** a) $x(x + 2)$ \- Multiply $x$ by $x$: $x \times x = x^2$ \- Multiply $x$ by $2$: $x \times 2 = 2x$ \- So, $x(x + 2) = x^2 + 2x$ b) $x(2x - 5)$ \- Multiply $x$ by $2x$: $x \times 2x = 2x^2$ \- Multiply $x$ by $-5$: $x \times (-5) = -5x$ \- So, $x(2x - 5) = 2x^2 - 5x$ c) $2x(3x + 4)$ \- Multiply $2x$ by $3x$: $2x \times 3x = 6x^2$ \- Multiply $2x$ by $4$: $2x \times 4 = 8x$ \- So, $2x(3x + 4) = 6x^2 + 8x$ d) $6x(x - 2y)$ \- Multiply $6x$ by $x$: $6x \times x = 6x^2$ \- Multiply $6x$ by $-2y$: $6x \times (-2y) = -12xy$ \- So, $6x(x - 2y) = 6x^2 - 12xy$ 4. **Summary:** $$\begin{aligned} a)\quad & x^2 + 2x \\ b)\quad & 2x^2 - 5x \\ c)\quad & 6x^2 + 8x \\ d)\quad & 6x^2 - 12xy \end{aligned}$$ Each expression is expanded by applying the distributive property carefully to each term inside the parentheses.