```Question 626189

Looking at the expression {{{b^4+2a^2b^2+a^4}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{2}}}, and the last coefficient is {{{1}}}.

Now multiply the first coefficient {{{1}}} by the last coefficient {{{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=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><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></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 {{{2a^2b^2}}} with {{{a^2b^2+a^2b^2}}}. Remember, {{{1}}} and {{{1}}} add to {{{2}}}. So this shows us that {{{a^2b^2+a^2b^2=2a^2b^2}}}.

{{{b^4+highlight(a^2b^2+a^2b^2)+a^4}}} Replace the second term {{{2a^2b^2}}} with {{{a^2b^2+a^2b^2}}}.

{{{(b^4+a^2b^2)+(a^2b^2+a^4)}}} Group the terms into two pairs.

{{{b^2(b^2+a^2)+(a^2b^2+a^4)}}} Factor out the GCF {{{b^2}}} from the first group.

{{{b^2(b^2+a^2)+a^2(b^2+a^2)}}} Factor out {{{a^2}}} 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.

{{{(b^2+a^2)(b^2+a^2)}}} Combine like terms. Or factor out the common term {{{b^2+a^2}}}

{{{(b^2+a^2)^2}}} Condense the terms.

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So {{{b^4+2a^2b^2+a^4}}} factors to {{{(b^2+a^2)^2}}}.

In other words, {{{b^4+2a^2b^2+a^4=(b^2+a^2)^2}}}.

Note: you can check the answer by expanding {{{(b^2+a^2)^2}}} to get {{{b^4+2a^2b^2+a^4}}} or by graphing the original expression and the answer (the two graphs should be identical).

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