How does the distributive property work with a negative factor?

The outside factor multiplies every term inside the parentheses. The negative sign belongs to that outside factor, so include it in each multiplication—not just the first one.

For example: −3(x + 4) = (−3)x + (−3)(4) = −3x − 12.

Both terms are positive inside, so multiplying each by −3 makes both products negative. Think of the operation as two separate multiplications. OpenStax illustrates the same pattern with −2(4y+1)=−8y−2. [S1]

Treat subtraction as adding a negative term. In x−4, the terms are x and −4. Then multiply both by the negative factor:

−3(x − 4) = (−3)x + (−3)(−4) = −3x + 12.

The second product is positive because a negative times a negative is positive. A common mistake is to copy the minus sign after distributing without multiplying it. You can also write the original as −3(x + (−4)); this makes the two products easier to see.

Try expanding −2(4y+1). Multiply −2 by each term: (−2)(4y)=−8y and (−2)(1)=−2. So the expanded expression is −8y−2. This is the same negative-factor example shown by OpenStax. [S1]

To check, choose a value such as y=3 and substitute it into both forms:

Original: −2(4·3+1) = −2(13) = −26. Expanded: −8·3−2 = −24−2 = −26.

The results agree for this input. That is a useful arithmetic check, though one matching input alone does not prove two expressions are identical. The distributive rule justifies the expansion; substitution helps catch calculation or sign errors. [S1]

Use this routine:

1. Keep the outside factor, including its sign, visible.
2. Write one multiplication for every term inside the parentheses.
3. Work out each product’s sign before simplifying.
4. Combine terms only after all products are written.
5. If unsure, substitute a convenient value into both forms and compare.

For instance, −2(4−3a) gives (−2)(4)+(−2)(−3a)=−8+6a. OpenStax shows the same sign pattern as −44+33a for −11(4−3a). [S1] Check with a=1: the original is −2(1)=−2, and the expanded form is −8+6=−2.

A copyable setup is:

−k(A − B) = (−k)A + (−k)(−B) = −kA + kB

Here k, A, and B stand for quantities. This layout highlights that the subtracted term is negative before multiplication. When terms include variables, multiply their coefficients and retain the variable part; for example, (−2)(4y)=−8y. A single substitution check does not establish equality by itself. [S1]