Key concept: If two expressions are equivalent, they must evaluate to the same value for every possible value of x.
For example, since the expression 2x + 3x is equivalent to the expression 5x, the two expressions will evaluate to the same number for every value of x.
So, if x = 7, the expression 2x + 3x = 2(7) + 3(7) = 14 + 21 = 35, and the expression 5x = 5(7) = 35

So for example, if \(x = 1\), our original expression becomes: \((1 - 2)^2 + (1 - 1)^2 + 1^2 + (1 + 1)^2 + (1 + 2)^2\), which evaluates to be 15.
So, the correct answer must also evaluate to be 15, when \(x = 1\).

So, we'll plug \(x = 1\) into each answer choice....
(A) \(5(1)^2 = 5\). ELIMINATE

(B) \(5(1)^2 + 10 =\) 15 Perfect!

(C) \((1)^2 + 10 = 11\). ELIMINATE

(D) \(5(1)^2 + 6(1) + 10 = 21\). ELIMINATE

(E) \(5(1)^2 - 6(1) + 10 = 9\). ELIMINATE

Answer: B

\((x - 2)^2 + (x - 1)^2 + x^2 + (x + 1)^2 + (x + 2)^2\)

In the above expression, we can clearly identify two pairs.

\((x-2)^2 \)and \((x+2)^2 \)

\((x-1)^2\) and \((x+1)^2\)

Each of the pair contains two expressions of the form \((a-b)^2\) and \((a+b)^2\)

When you distribute and add them both as demanded by the expression, the \(2ab\) term will cancel out, leaving behind \(2a^2 + 2b^2\)

You can verify this by distributing by FOILing

\(a^2 + b^2 -2ab + a^2 + b^2 + 2ab = 2a^2 + 2b^2\)

So the first pair gives us \(2x^2 + 2.2^2 = 2x^2 + 8\)

The second pair gives us

\(2x^ + 2.1^2 = 2x^2 + 2\)

When we add them both we get \(4x^2 + 10.\)

But there is also an \(x^2\) left in the expression which needs to be added as well. This gives us

\(5x^2 + 10.\)