Question 96009
factor 
<pre><font size = 4><b>
<font color = "red">2</font>x² + 9xy + <font color = "red">10</font>y²

1. Multiply the <font color = "red">2</font> by the <font color = "red">10</font>, get 20.

2. Notice the sign of the last term is +, so we think "SUM".
(If the sign of the last term had been -, we would think "DIFFERENCE")

3. Think of two positive integers which have product 20 and SUM 9, the
coefficient (in absolute value) of the middle term.

It doesn't take long to see that two such positive integers are 5 and 4.

4. Now use 5 and 4 to rewrite the 9 as (5 + 4)

2x² + (5 + 4)xy + 10y²

5. Remove the parentheses by distributing

2x² + 5xy + 4xy + 10y²

6. Factor by grouping. That is,

a. factor the first two terms by taking
out x.

b. factor the last two terms by taking out 2y

x(2x + 5y) + 2y(2x + 5y)

c. Now notice that <font color = "red">(2x + 5y)</font> is contained in both expressions:

x<font color = "red">(2x + 5y)</font> + 2y<font color = "red">(2x + 5y)</font>

d. So factor out the whole <font color = "red">(2x + 5y)</font> leaving x when factoring
it out of the left expression, and leaving 2y when factoring
<font color = "red">(2x + 5y)</font> out on the right expression.

<font color = "red">(2x + 5y)</font>(x + 2y)

That's it!

Edwin</pre>