To determine which expression is equivalent to [tex]\((r(x))^2 - 2(s(x))\)[/tex], let's start by evaluating [tex]\(r(x)\)[/tex] and [tex]\(s(x)\)[/tex] for a given value of [tex]\(x\)[/tex].
Given:
[tex]\[ r(x) = 2x + 2 \][/tex]
[tex]\[ s(x) = x^2 + 4x \][/tex]
First, we calculate [tex]\((r(x))^2\)[/tex]:
[tex]\[ r(x) = 2x + 2 \][/tex]
[tex]\[ (r(x))^2 = (2x + 2)^2 \][/tex]
Using the binomial expansion formula [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex], we get:
[tex]\[ (2x + 2)^2 = (2x)^2 + 2(2x)(2) + 2^2 \][/tex]
[tex]\[ = 4x^2 + 8x + 4 \][/tex]
Next, we calculate [tex]\(2(s(x))\)[/tex]:
[tex]\[ s(x) = x^2 + 4x \][/tex]
[tex]\[ 2(s(x)) = 2(x^2 + 4x) \][/tex]
[tex]\[ = 2x^2 + 8x \][/tex]
Now, we substitute these results into the expression [tex]\((r(x))^2 - 2(s(x))\)[/tex]:
[tex]\[ (r(x))^2 - 2(s(x)) = (4x^2 + 8x + 4) - (2x^2 + 8x) \][/tex]
Subtracting the two expressions, we obtain:
[tex]\[ 4x^2 + 8x + 4 - 2x^2 - 8x \][/tex]
Combine like terms:
[tex]\[ 4x^2 - 2x^2 + 8x - 8x + 4 \][/tex]
[tex]\[ = 2x^2 + 4 \][/tex]
Therefore, the expression equivalent to [tex]\((r(x))^2 - 2(s(x))\)[/tex] is:
[tex]\[ \boxed{2x^2 + 4} \][/tex]
