To simplify the expression [tex]\( x^2 - 4(a - b)^2 \)[/tex], let’s go through the step-by-step process:
1. Identify the given expression: [tex]\( x^2 - 4(a - b)^2 \)[/tex].
2. Notice the form of the second term: The term [tex]\( 4(a - b)^2 \)[/tex] is a square of a binomial times a constant (4 in this case).
3. Recall the basic algebraic identity for the square of a binomial:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
4. Multiply the identity by 4 to express [tex]\( 4(a - b)^2 \)[/tex]:
[tex]\[
4(a - b)^2 = 4(a^2 - 2ab + b^2)
\][/tex]
Therefore,
[tex]\[
4(a - b)^2 = 4a^2 - 8ab + 4b^2
\][/tex]
5. Substitute this back into the original expression: Replace [tex]\( 4(a - b)^2 \)[/tex] with [tex]\( 4a^2 - 8ab + 4b^2 \)[/tex]:
[tex]\[
x^2 - 4(a - b)^2 = x^2 - (4a^2 - 8ab + 4b^2)
\][/tex]
6. Distribute the negative sign: Apply the negative sign to each term within the parenthesis:
[tex]\[
x^2 - 4a^2 + 8ab - 4b^2
\][/tex]
7. Combine all the terms together: List all terms in the expression:
[tex]\[
x^2 - 4a^2 + 8ab - 4b^2
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
x^2 - 4a^2 + 8ab - 4b^2
\][/tex]
