Factorize fully 𝑥 plus four 𝑦 plus three squared minus 𝑥 minus four 𝑦 minus three squared.
The key here is to notice that one square term is being subtracted from another square term. And we call this the difference of squares. 𝑎 squared minus 𝑏 squared is equal to 𝑎 plus 𝑏 times 𝑎 minus 𝑏. We can use this to help us factorize.
For us, 𝑎 equals 𝑥 plus four 𝑦 plus three. And 𝑏 equals 𝑥 minus four 𝑦 minus three. We need 𝑎 plus 𝑏 times 𝑎 minus 𝑏, which would look like this. The key here is to be really careful with our positive and negative signs, 𝑥 plus four 𝑦 plus three plus 𝑥 minus four 𝑦 minus three and then 𝑥 plus four 𝑦 plus three. This time, we’re subtracting. So we need to distribute the subtraction. Minus 𝑥 minus negative four 𝑦 equals plus four 𝑦. And minus negative three equals plus three.
The next step will be to combine like terms. 𝑥 plus 𝑥 equals two 𝑥. Positive four 𝑦 minus four 𝑦 cancels out. Positive three minus three also cancels out. And that means our 𝑎 plus 𝑏 term has been simplified to two 𝑥.
On the other side, we have the 𝑎 minus 𝑏 terms. Positive 𝑥 minus 𝑥 cancels out. Four 𝑦 plus four 𝑦 equals eight 𝑦. And three plus three is six. Our 𝑎 minus 𝑏 terms equals eight 𝑦 plus six. But eight 𝑦 and six share a common factor. They both have a factor of two. If we take out that common factor of two, we could rewrite this to say, two times four 𝑦 plus three.
And remember, we’re multiplying the 𝑎 plus 𝑏 times the 𝑎 minus 𝑏. We would then have two 𝑥 times two times four 𝑦 plus three. We can multiply this, two times two, which equals four. Bring down the 𝑥. And then bring down the four 𝑦 plus three.
The fully factorized form is four 𝑥 times four 𝑦 plus three.