```Question 352211

{{{5(4w^2+20wg+25g^2)}}} Factor out the GCF {{{5}}}

Now let's focus on the inner expression {{{4w^2+20wg+25g^2}}}

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Looking at {{{4w^2+20wg+25g^2}}} we can see that the first term is {{{4w^2}}} and the last term is {{{25g^2}}} where the coefficients are 4 and 25 respectively.

Now multiply the first coefficient 4 and the last coefficient 25 to get 100. Now what two numbers multiply to 100 and add to the  middle coefficient 20? Let's list all of the factors of 100:

Factors of 100:

1,2,4,5,10,20,25,50

-1,-2,-4,-5,-10,-20,-25,-50 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 100

1*100

2*50

4*25

5*20

10*10

(-1)*(-100)

(-2)*(-50)

(-4)*(-25)

(-5)*(-20)

(-10)*(-10)

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to 20? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 20

<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">100</td><td>1+100=101</td></tr><tr><td align="center">2</td><td align="center">50</td><td>2+50=52</td></tr><tr><td align="center">4</td><td align="center">25</td><td>4+25=29</td></tr><tr><td align="center">5</td><td align="center">20</td><td>5+20=25</td></tr><tr><td align="center">10</td><td align="center">10</td><td>10+10=20</td></tr><tr><td align="center">-1</td><td align="center">-100</td><td>-1+(-100)=-101</td></tr><tr><td align="center">-2</td><td align="center">-50</td><td>-2+(-50)=-52</td></tr><tr><td align="center">-4</td><td align="center">-25</td><td>-4+(-25)=-29</td></tr><tr><td align="center">-5</td><td align="center">-20</td><td>-5+(-20)=-25</td></tr><tr><td align="center">-10</td><td align="center">-10</td><td>-10+(-10)=-20</td></tr></table>

From this list we can see that 10 and 10 add up to 20 and multiply to 100

Now looking at the expression {{{4w^2+20wg+25g^2}}}, replace {{{20wg}}} with {{{10wg+10wg}}} (notice {{{10wg+10wg}}} adds up to {{{20wg}}}. So it is equivalent to {{{20wg}}})

{{{4w^2+highlight(10wg+10wg)+25g^2}}}

Now let's factor {{{4w^2+10wg+10wg+25g^2}}} by grouping:

{{{(4w^2+10wg)+(10wg+25g^2)}}} Group like terms

{{{2w(2w+5g)+5g(2w+5g)}}} Factor out the GCF of {{{2w}}} out of the first group. Factor out the GCF of {{{5g}}} out of the second group

{{{(2w+5g)(2w+5g)}}} Since we have a common term of {{{2w+5g}}}, we can combine like terms

So {{{4w^2+10wg+10wg+25g^2}}} factors to {{{(2w+5g)(2w+5g)}}}

So this also means that {{{4w^2+20wg+25g^2}}} factors to {{{(2w+5g)(2w+5g)}}} (since {{{4w^2+20wg+25g^2}}} is equivalent to {{{4w^2+10wg+10wg+25g^2}}})

note:  {{{(2w+5g)(2w+5g)}}} is equivalent to  {{{(2w+5g)^2}}} since the term {{{2w+5g}}} occurs twice. So {{{4w^2+20wg+25g^2}}} also factors to {{{(2w+5g)^2}}}

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So our expression goes from {{{5(4w^2+20wg+25g^2)}}} and factors further to {{{5(2w+5g)^2}}}

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So {{{20w^2+100wg+125g^2}}} factors to {{{5(2w+5g)^2}}}

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