Simon's factoring trick(complete the rectangle)

 "Complete the Rectangle", also called Simon’s Favorite Factoring Trick, is a clever algebraic method for factoring expressions of the form:

$$xy + ax + by + c$$

Or more commonly, you'll see it used in a simpler form:

$$xy + ax + by + d$$

But especially when we’re given something like:

$$xy + ax + by + ab$$

It becomes very easy to factor. Let me walk you through it step-by-step.

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💡 The Key Idea

We add and subtract a constant to turn the expression into a perfect rectangle (a.k.a. a factorable quadratic or product of binomials).

The “complete the rectangle” version of this trick usually works best on expressions like:

$$xy + ax + by + ab$$

We treat it like this:

$$xy + ax + by + ab = (x + b)(y + a)$$

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✅ Step-by-Step Example

Factor:

$$xy + 3x + 2y + 6$$

Step 1: Rearrange the terms:

Group like this:

$$xy + 3x + 2y + 6$$

We’re hoping to match the pattern:

$$xy + ax + by + ab$$

So what values of $a$ and $b$ would give the last term as $ab = 6$?
Let’s try $a = 3$, $b = 2$, since $3 \cdot 2 = 6$.

Now check:

$$xy + 3x + 2y + 6 = xy + 3x + 2y + 3 \cdot 2$$

So now we use Simon's Trick:

$$xy + 3x + 2y + 6 = (x + 2)(y + 3)$$

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💡 Why It Works

Because:

$$(x + b)(y + a) = xy + ax + by + ab$$

So when you see:

$$xy + ax + by + ab$$

You can jump straight to:

$$(x + b)(y + a)$$

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🔧 Practice Problem

Try factoring this using Simon’s Trick:

$$xy + 4x + 5y + 20$$

Solution:

Check: $4x = ax$, $5y = by$, and $ab = 4 \cdot 5 = 20$

So:

$$xy + 4x + 5y + 20 = (x + 5)(y + 4)$$

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🧠 Summary

Simon’s Favorite Factoring Trick lets you factor expressions of the form:

$$xy + ax + by + ab = (x + b)(y + a)$$

Look for:

• A term $xy$,

• Linear terms in $x$ and $y$,

• A constant that is the product of the coefficients of $x$ and $y$.

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Now, let's extend it further for the case when the coefficient of xy is not 1.
For e.g.
5xy - 66x - 66y = -1
In such cases we will multiply the entire equation with the coefficient of xy:
25xy - 5.66x - 5.66y = -5
5x(5y - 66) - 66(5y) = -5
Now again add and subtract the original coefficients of x,y(-66,-66).
5x(5y - 66) - 66(5y - 66) + -66*66 = -5
(5x - 66)(5y - 66) = 66^2 - 5 = 4351