We will solve this problem by finding factor pairs. Factor pairs are composed of two numbers that are multiplied together to equal a product. List all the factor pairs of Sam’s gummy bears.

\(\displaystyle 1\times35=35\)

\(\displaystyle 5\times7=35\)

Do not forget to list their reciprocals.

\(\displaystyle 7\times5=35\)

\(\displaystyle 35\times1=35\) 

Sam can make \(\displaystyle 4\) different gift bag combinations with an even amount of gummy bears in each bag.

Simplify the following expression:

\(\displaystyle 4x(2x+x^3)\)

When we are multiplying something in parentheses, we need to use the distributive property and multiply each term within the parentheses. With that in mind we can rewrite the given expression

\(\displaystyle 4x(2x+x^3)=(4x*2x)+(4x*x^3)\)

Next, we need to do the multiplication. Recall that when multiplying variables, we add the exponents together and multiply the coefficients:

\(\displaystyle (4x*2x)+(4x*x^3)=8x^2+4x^4\)

So our answer is:

\(\displaystyle 8x^2+4x^4\)

To distribute a value into parantheses, you must multiply all the separate values by the value on the outside.  

So 

\(\displaystyle 3*x=3x\) 

and 

\(\displaystyle 3*2y=6y\) 

resulting in an answer of 

\(\displaystyle 3x+6y\).

Using the distributive property, you would distribute what is outside of the brackets or the parentheses to whatever is inside the parentheses.

\(\displaystyle (3 \times 6) + (3 \times 2) = 18 + 6 = 24\)

\(\displaystyle (2 \times 8) + (2 \times 1) = 16 + 2 = 18\)

Now take the difference of the two values found above.

\(\displaystyle 24 - 18 = 6\)

To solve using the distributive property, we take the number on the outside and multiply it by each term on the inside of the parentheses.  

So in the problem,

\(\displaystyle 5(x+4)\)

we will multiply or distribute the 5 through.

We will multiply the 5 by the first term.  We get

\(\displaystyle {\color{Green} 5}({\color{Green} x}+4) \rightarrow {\color{Green} 5x}\)

Now, we multiply the 5 by the second term.  We get

 \(\displaystyle {\color{Red} 5}(x{\color{Red} +4}) \rightarrow {\color{Red} +20}\)

Now, we combine the two answers.  We get

\(\displaystyle {\color{Green} 5x} {\color{Red} +20}\)

or

\(\displaystyle 5x+20\)

With the distributive property if there is a common factor in the terms of an expression, you can take it out of all of them a place it outside of parenthesis around the expression.  

Since \(\displaystyle 3x\) and \(\displaystyle 15\) have a common factor of \(\displaystyle 3\), you can divide both by \(\displaystyle 3.\) 

This would leave you with your answer of 

\(\displaystyle 3(x+5)\).

With the distributive property, you want to multiply both numbers inside the parentheses by 4.

Therefore, you are left with

\(\displaystyle 4*3 + 4*2=x\).

Then we have,

\(\displaystyle 12 + 8=20\).

With the distributive property you must multiply each number in the parentheses by the number outside of the parentheses.

So to start the problem, we have

\(\displaystyle 2*3 + 2*1 + 3*2 + 3*3.\) 

Then we simplify this to

\(\displaystyle 6+2+6+9=x\).

So,

\(\displaystyle x=23\).

To start the problem, first multiply what is inside the parentheses by what is outside the parentheses.

So, we first have

\(\displaystyle 2x+2*4=48\).

Then,

\(\displaystyle 2x+8=48\).

Then we must get \(\displaystyle x\) alone, so we subtract 8 from both sides of the equation and are left with

\(\displaystyle 2x=40\).

To get \(\displaystyle x\) alone, we need to divide both sides by 2, so we are left with 

\(\displaystyle x=20.\)

To use the distributive property, you multiply the outside value by each one in the parentheses.  

\(\displaystyle 3x*x=3x^{2}\) 

and 

\(\displaystyle 3x*4=12x\) 

so you answer is 

\(\displaystyle 3x^{2}+12x\).