Combining the Perfect Square Formulas

Let's recall the square of binomials formulas

Product of the sum and difference of two terms
$(X + Y)\cdot(X - Y) = X^2 - Y^2$
You can use this formula when there is a product between the sum of two specific terms and the difference of those same terms.
Instead of showing them as a product of sum and difference, you can write $X^2 - Y^2$
This rule also works in reverse.

The Square of a Difference Formula
$(X - Y)^2=X^2 - 2XY + Y^2$
When we encounter two numbers with a minus sign between them - meaning a difference, and they are wrapped in parentheses and raised as one expression to the second power, we can use this formula.

The Square of a Sum Formula
$(X + Y)^2=X^2+ 2XY + Y^2$
- When encountering two numbers with a plus between them - meaning a sum, and they are both enclosed in parentheses and raised as one expression to the second power, we can use this formula.

Formulas relating to two expressions to the power of 3
$(a+b)^3=a^3+3a^2 b+3ab^2+b^3$
A way to express the sum of two terms, when they are enclosed in parentheses and raised as one expression to the power of three.
$(a-b)^3=a^3-3a^2 b+3ab^2-b^3$
A way to express the difference of two terms, when they are enclosed in parentheses and raised as one expression to the power of three.

And now? Let's practice exercises that combine several perfect square formulas together!
Ready?
Here's an exercise:
Solve the following exercise using the perfect square formulas -
$x^2+4+(5x+1)^2=(4x-1)^2+(x+2)\cdot (x-2)$

Solution:
Although the exercise looks complex, if we solve it step by step using the square formulas, you'll see it's simple and easy to solve.
Let's start reading the exercise from left to right and understand which expression can match a square formula.
$(5x+1)^2$
This expression matches the square of sum formula -
The square of sum formula
$(X + Y)^2=X^2+ 2XY + Y^2$
Therefore
$​​​​​​​(5x+1)^2=5^2+2\cdot 5x\cdot 1+1^2$
$(5x+1)^2=25x^2+10x+1$
Let's write this in the original exercise and we obtain:
$x^2+4+25x^2+10x+1=(4x-1)^2+(x+2)\cdot (x-2)$
Let's continue reading the exercise... aha
The expression
$(4x-1)^2$
matches the square of difference formula
The square of difference formula
$(X - Y)^2=X^2 - 2XY + Y^2$
Therefore
$(4x-1)^2=16x^2-8x+1$
Let's write this in the original exercise after we've dealt with it and obtain:
$x^2+4+25x^2+10x+1=16x^2-8x+1+(x+2)\cdot (x-2)$
Let's continue reading the exercise... aha
The expression
$(x+2)\cdot (x-2)$
matches the formula for the product of sum and difference
$(X + Y)\cdot(X - Y) = X^2 - Y^2$
Therefore we can replace
$(x+2)\cdot (x-2)$
with$x^2-2^2$
Let's write this in the original exercise after we've dealt with it:
$x^2+4+25x^2+10x+1=16x^2-8x+1+ x^2-2^2$

Note - we can write $2^2=4$ and obtain the following:
$x^2+4+25x^2+10x+1=16x^2-8x+1+ x^2-4$
Excellent! Notice that we can cancel $x^2-4$ from both sides and obtain:
$25x^2+10x+1=16x^2-8x+1$
Move terms to one side and obtain:
$9x^2+18x=0$
Factor out:
$9x(x^2+2)=0$
For the equation to equal zero, we can substitute $0$ once
because $9x=0$
$x=0$
And a second solution:
$(x^2+2)=0$
This expression will never equal zero because $x^2$ will never be negative, so there is only one solution which is $x=0$