Expand and simplify: (a) (a+4)(3a+7) (b) (x+10)(x-10) (c) (2x-5y)(3x-4y)

$$3x(x - 2y - 5z) + x(x + 6y)$$ $$3x \cdot x - 3x \cdot 2y - 3x \cdot 5z + x \cdot x + x \cdot 6y$$ $$3x^2 - 6xy - 15xz + x^2 + 6xy$$

$$(3x^2 + x^2) + (-6xy + 6xy) - 15xz$$ $$4x^2 + 0 - 15xz$$ $$4x^2 - 15xz$$

The expanded and simplified expression is \boxed{\text{4x^2 - 15xz}}.

4. a)

$$(a)(3a) + (a)(7) + (4)(3a) + (4)(7)$$ $$3a^2 + 7a + 12a + 28$$

$$3a^2 + (7a + 12a) + 28$$ $$3a^2 + 19a + 28$$

The expanded and simplified expression is \boxed{\text{3a^2 + 19a + 28}}.

b)

This is also a difference of squares pattern $(A+B)(A-B) = A^2 - B^2$.

$$(x)(x) + (x)(-10) + (10)(x) + (10)(-10)$$ $$x^2 - 10x + 10x - 100$$

$$x^2 + (-10x + 10x) - 100$$ $$x^2 + 0 - 100$$ $$x^2 - 100$$

The expanded and simplified expression is \boxed{\text{x^2 - 100}}.

c)

$$(2x)(3x) + (2x)(-4y) + (-5y)(3x) + (-5y)(-4y)$$ $$6x^2 - 8xy - 15xy + 20y^2$$

$$6x^2 + (-8xy - 15xy) + 20y^2$$ $$6x^2 - 23xy + 20y^2$$

The expanded and simplified expression is \boxed{\text{6x^2 - 23xy + 20y^2}}.