Simplify the following algebraic expressions.

$$2x \times x^2 = 2 \cdot x^{1+2}$$ $$= 2x^3$$

The simplified expression is $\boxed{\mathbf{2x^3}}$.

2. $4y^3 \times 3y$

$$= 12y^{3+1}$$ $$= 12y^4$$

The simplified expression is $\boxed{\mathbf{12y^4}}$.

3. $2a^2 \times 3a$

$$= 6a^{2+1}$$ $$= 6a^3$$

The simplified expression is $\boxed{\mathbf{6a^3}}$.

4. $3b \times 5b$

$$= 15b^{1+1}$$ $$= 15b^2$$

The simplified expression is $\boxed{\mathbf{15b^2}}$.

5. $6x^2 \times 7x^3$

$$= 42x^{2+3}$$ $$= 42x^5$$

The simplified expression is $\boxed{\mathbf{42x^5}}$.

Classwork: Simplify the following

1. $3(x-y)$

$$3(x-y) = 3 \times x - 3 \times y$$ $$= 3x - 3y$$

The simplified expression is $\boxed{\mathbf{3x - 3y}}$.

2. $4(m+1)$

$$4(m+1) = 4 \times m + 4 \times 1$$ $$= 4m + 4$$

The simplified expression is $\boxed{\mathbf{4m + 4}}$.

3. $2(2a+3b)$

$$2(2a+3b) = 2 \times 2a + 2 \times 3b$$ $$= 4a + 6b$$

The simplified expression is $\boxed{\mathbf{4a + 6b}}$.

4. $5(c-4)$

$$5(c-4) = 5 \times c - 5 \times 4$$ $$= 5c - 20$$

The simplified expression is $\boxed{\mathbf{5c - 20}}$.

5. $6(2x+3-1)$

$$= 6 \times 2x + 6 \times 2$$ $$= 12x + 12$$

The simplified expression is $\boxed{\mathbf{12x + 12}}$.