Answer: [tex](x+2)^2+3\\\\[/tex]
a = 2 and b = 3
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Explanation:
Let's expand out [tex](x+a)^2+b[/tex] to get the following:
[tex](x+a)^2+b\\\\(x+a)(x+a)+b\\\\x(x+a)+a(x+a)+b\\\\x^2+ax+ax+a^2+b\\\\x^2+2ax+a^2+b\\\\[/tex]
The x term here is 2ax
Compare this to the x term of [tex]x^2+4x+7[/tex] and we see that
[tex]2ax = 4x\\\\2a = 4\\\\a = 4/2\\\\a = 2\\\\[/tex]
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The constant term of [tex]x^2+2ax+a^2+b\\\\[/tex] is the [tex]a^2+b[/tex] portion since it doesn't have the variable x attached to it.
Compare this with the 7 of [tex]x^2+4x+7[/tex] which is also the constant.
Equate the two items, plug in a = 2 and solve for b.
[tex]a^2+b = 7\\\\2^2+b = 7\\\\4+b = 7\\\\b = 7-4\\\\b = 3\\\\[/tex]
Therefore,
[tex]x^2+4x+7 = (x+a)^2+b\\\\x^2+4x+7 = (x+2)^2+3\\\\[/tex]
