Question

What is a factor of \[4 - {(x + y)^2}\] and why?
A) \[ - {(x + y)^2}\]
B) \[x + y\]
C) \[2 - x + y\]
D) \[2 + x + y\]
E) \[4 + x + y\]

Answer 1

Hint: Factors are like small pieces which combine together to get a big piece or we can say that these are simple expressions which are multiplied to get a complete expression.
\[(x + \alpha )(x + \alpha ) \to {x^2} + 2\alpha x + {\alpha ^2}\]
Here, $x + \alpha $ and $x + \alpha $are the factors for the algebraic expression ${x^2} + 2x\alpha + {\alpha ^2}$ .$$
Formula used: \[{a^2} - {b^2} = (a - b)(a + b)\]

Complete step-by-step solution:
Step 1: We can write the given algebraic expression in form of,
\[4 - {(x + y)^2}\]\[ = {2^2} - {(x + y)^2}\]
Step 2: Now comparing both equation i.e.
\[{a^2} - {b^2} = (a - b)(a + b)\].................. Equation-1
\[4 - {(x + y)^2}\]\[ = {2^2} - {(x + y)^2}\].............. Equation-2
\[{a^2} - {b^2} = {2^2} - {(x + y)^2}\]
Comparing first term of both sides, we get
\[{a^2} = {2^2}\]
\[a = 2\]
Now, comparing second term, we get
\[ - {b^2} = - {(x + y)^2}\]
Multiplying \[( - 1)\] on both sides, we get
\[{b^2} = {(x + y)^2}\]
\[b = (x + y)\]
We get ,\[a = 2\] & \[b = x + y\]
Step 3: By using equation-1, we can write,
\[4 - {(x + y)^2}\]\[ = {2^2} - {(x + y)^2}\]
\[\Rightarrow 4 - {(x + y)^2}\]\[ = (2 - (x + y))(2 + (x + y))\]
\[\Rightarrow 4 - {(x + y)^2}\] \[ = (2 - x - y)(2 + x + y)\]
Hence, \[(2 - x - y)\]&\[(2 + x + y)\] are the two factors of the given expression which multiplied with each other to give actual expression.
In given options , we only have\[(2 + x + y)\], so correct option is (D)
Options (A) (B) (C) & (E) cannot be correct because if we multiply them with another factor then we do not get the desired expression that is\[4 - {(x + y)^2}\]. So only option (D) is correct.
Additional information: Here is a list of expressions which make factorization easier

Expand	Factor
$a^2-b^2$	$\left(a+b\right) \left(a-b\right)$
$a^2+2ab+b^2$	$\left(a+b\right) \left(a+b\right)$
$a^2-2ab+b^2$	$\left(a-b\right) \left(a-b\right)$
$a^3+b^3$	$\left(a+b\right) \left(a^2-ab+b^2\right)$
$a^3-b^3$	$\left(a-b\right) \left(a^2+ab+b^2\right)$

Note: During factorization, follow these steps one by one:
> Factor out any common terms from the expression.
> See if it fits any of the expressions given above and others.
> Keep going until factorization cannot be done further.
> To check the answer we can multiply each factor with each other, if we get the same expression then the calculated answer is correct otherwise not.