Regarding the binomial formula

Question

$(a+b)(a-b) = a^2 - b^2$

so if I have:

$-(a^2 + b^2)=$

$= -a^2 - b^2$

Can I write it like:

$(-a^2) - b^2 = $

$= (-a + b)(-a - b)$

Answer 1

Be careful: $-a^2$ is “the negative of the square of $a$”, not “the square of the negative of $a$”. The latter would be written $(-a)^2$.

Exponents have higher priority over operations based on addition (such as taking the negative). Thus $-a^2$ is computed by first taking the square of $a$, then the negative.

By the way, $(-a)^2=a^2$, which equals $-a^2$ if and only if $a=0$.

Also, notice that $$ (-a+b)(-a-b)=(-1)(a-b)(-1)(a+b)=(a-b)(a+b)=a^2-b^2 $$ that can be obtained also with the standard $$ (-a+b)(-a-b)=(-a)^2-b^2=a^2-b^2 $$

Answer 2

The formula is $(a+b)(a-b)=a^2-b^2$, therefore $$(-a+b)(-a-b)=(-a)^2-b^2=a^2-b^2\neq -a^2-b^2$$