I'll show you how to do this with a detailed example:
Example: Factor x² + 4x - 32 = 0.
To solve this question, I start by making a box that will reveal the answer.
I start by writing the "squared" term in the upper left and the last term (the constant term) in the lower right:
I am going to write the factors on the top and the left sides. This is just a table - you multiply the things at the top by the things on the left and put the product inside the corresponding box. First, I know that x² is just x times x:
Now comes the hard part. I need to find two numbers that add to +4 (that comes from the +4x middle term) and that multiply to -32.
So I start by ignoring the minus sign for a moment and just think about what pairs of numbers multiply to make 32.
I'll do it the long way so you can see. Each pair, I try switching the signs (one is positive and the other negative) and add them up. I'm hoping to add to 4.
-1 + 32 = 31 (nope)
This means the two numbers I'm looking for are -4 and 8. I write them on the diagram. Note that where the column and row meet, I have -32, which is +8 times -4.
Now I can fill in the two empty spots. The row "x" times the column "+8" is +8x:
And the row "-4" times the column "x" is -4x:
If you were to add up everything inside the four cells, you should get the original quadratic (once you simplify): x² + (+8x) + (-4x) + (-32) = x² + 4x - 32.
The box I worked with reveals the factors of the quadratic: they are the things outside the cells!
So the factors of x² + 4x - 32 are (x + 8) and (x - 4).
In other words, x² + 4x - 32 = (x + 8) (x - 4).
Try this method with your question, Rebecca. The trick will be to find a pair of numbers that multiplies to make +10 and that adds to make +7.
Hope this helps,
Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.