Distributive property

The rule

$$a(b + c) = ab + ac$$

Every term inside gets multiplied:

$$4(x + 3) = 4x + 12$$

Why it is true

Read $4(x + 3)$ as four copies of $(x + 3)$:

$$(x+3) + (x+3) + (x+3) + (x+3)$$

Collecting the pieces gives four $x$s and four $3$s, which is $4x + 12$. The property is a shortcut for that counting, not a separate rule to memorise.

Watch the sign

A negative multiplier changes the sign of everything it reaches:

$$-2(x - 5) = -2x + 10$$

The second product is $-2 \times -5 = +10$. Dropping that sign change is the most common error in multi-step equations.

A lone minus sign distributes too

$-(x - 6)$ means $-1(x - 6)$, which is $-x + 6$. An invisible $-1$ is still a multiplier.