Final answer:
To factor the perfect square trinomial x²+20x+100, we can rewrite it as (x+10)² by completing the square.
Step-by-step explanation:
To factor a perfect square trinomial, we need to rewrite it in the form (x-a)², where 'a' is a constant. In the given equation, x²+20x+100, we can see that the first term x² is already in the form (x-a)², and the last term 100 is a perfect square (10²). So, we can rewrite the equation as (x+10)². However, the middle term 20x is not twice the product of x and a constant, so we need to manipulate the equation to make it fit the perfect square trinomial form.
To complete the square, we divide the middle term coefficient by 2, square it, and add it to both sides of the equation. In this case, we have x²+20x+100 = 64. The middle term coefficient is 20, so we divide it by 2 to get 10. Squaring 10 gives us 100, so we add 100 to both sides of the equation:
x²+20x+100+100 = 64+100
x²+20x+200 = 164
Now we can factor the perfect square trinomial (x²+20x+200) by rewriting it as (x+10)². Therefore, the correct answer is: (x+10)².
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