Combining like terms with distribution

We use Please Excuse My Dear Aunt Sally (P E MD AS) to remember the order of operations when simplifying an expression. According to this acronym, we perform calculations within the parentheses first. However if we encounter a problem where we are asked to simplify an expression such as $$5(2x + 3) + 4$$, we do not have like terms to combine within the parentheses, nor are there any exponents, so we perform the multiplication first. In this case, the 5 is multiplied by everything in the parentheses, which is called using the distributive property. We multiply 5 by 2x and 5 by 3. The problem then becomes:

$$10x + 15 + 4$$, note that the parentheses can be dropped because all of the operations in the expression are the same.

Now we can perform the addition of like terms to get:

$$10x + 19$$

Example 1
$$3(4m +6) + 5m +2$$

There are no like terms to combine within the parentheses P. There are no exponents E.  There is multiplication but no division  MD, so we multiply through the parentheses by 3.

$$\begin{align} & =12m + 18 + 5m + 2 \qquad \text{Multiply through the parentheses by 3}\\ & =17m + 20 \qquad \text{Add, or combine, like terms, the two m terms and the two number terms}\\ \end{align}$$

Example 2
$$3(8y + 4) + 2y + 9(3y + 5)$$

Again there are no like terms to combine within the parentheses, but there are now two sets of parentheses. Multiply through the first set by 3 and through the second set by 9.

$$\begin{align} & =24y + 12 + 2y + 27y + 45 \qquad \text{Multiply through the parentheses by 3 and 9}\\ & =53y + 57 \qquad \text{Combine like terms, the two y terms and the two number terms}\\ \end{align}$$

Example 3
$$-2(7x – 4) + 6x + 5(3x – 2) + 12$$

In this problem, we begin by multiplying through the first set of parentheses with a negative 2 and through the second set of parentheses by 5.

$$\begin{align} & =-14x + 8 + 6x + 15x – 10 + 12 \qquad \text{Multiply through the parentheses by -2 and 5}\\ & =7x + 10 \qquad \text{Combine like terms}\\ \end{align}$$