```Question 238740
Let {{{z=a+4}}}. So the expression {{{(a+4)^2 - 2(a+4) +1}}} then becomes {{{z^2-2z+1}}}

Looking at the expression {{{z^2-2z+1}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-2}}}, and the last term is {{{1}}}.

Now multiply the first coefficient {{{1}}} by the last term {{{1}}} to get {{{(1)(1)=1}}}.

Now the question is: what two whole numbers multiply to {{{1}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-2}}}?

To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{1}}} (the previous product).

Factors of {{{1}}}:

1

-1

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to {{{1}}}.

1*1 = 1
(-1)*(-1) = 1

Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-2}}}:

<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>1+1=2</font></td></tr><tr><td  align="center"><font color=red>-1</font></td><td  align="center"><font color=red>-1</font></td><td  align="center"><font color=red>-1+(-1)=-2</font></td></tr></table>

From the table, we can see that the two numbers {{{-1}}} and {{{-1}}} add to {{{-2}}} (the middle coefficient).

So the two numbers {{{-1}}} and {{{-1}}} both multiply to {{{1}}} <font size=4><b>and</b></font> add to {{{-2}}}

Now replace the middle term {{{-2z}}} with {{{-z-z}}}. Remember, {{{-1}}} and {{{-1}}} add to {{{-2}}}. So this shows us that {{{-z-z=-2z}}}.

{{{z^2+highlight(-z-z)+1}}} Replace the second term {{{-2z}}} with {{{-z-z}}}.

{{{(z^2-z)+(-z+1)}}} Group the terms into two pairs.

{{{z(z-1)+(-z+1)}}} Factor out the GCF {{{z}}} from the first group.

{{{z(z-1)-1(z-1)}}} Factor out {{{1}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

{{{(z-1)(z-1)}}} Combine like terms. Or factor out the common term {{{z-1}}}

{{{(z-1)^2}}} Condense the terms.

So {{{z^2-2z+1}}} factors to {{{(z-1)^2}}}.

In other words, {{{z^2-2z+1=(z-1)^2}}}.

Now plug in {{{z=a+4}}} to go from {{{(z-1)^2}}} to {{{(a+4-1)^2}}}. Now simplify to get {{{(a+3)^2}}}

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