Final answer:
The expressions are not perfect squares. A(x) = x^2 + 6x + 13 is not factorable into the form (a + b)^2. A(x) = 2x^2 + 8x + 14 can be rewritten as 2(x + 2)^2 + 3, which is the perfect square closest to the given expression.
Step-by-step explanation:
To determine if each expression is a perfect square, we need to check if it can be factored into the form (a + b)^2. Let's look at the first expression, A(x) = x^2 + 6x + 13.
This expression cannot be factored into the form (a + b)^2, so it is not a perfect square.
Next, we have A(x) = 2x^2 + 8x + 14. Again, this expression cannot be factored into the form (a + b)^2, so it is not a perfect square. To find the perfect square closest to this expression, we can rewrite it as 2(x^2 + 4x + 7).
Now, let's complete the square for x^2 + 4x + 7. First, we half the coefficient of x and square it, which gives us (4/2)^2 = 4. We add 4 inside the parentheses and subtract 4 * 2 outside the parentheses to keep the expression equivalent:
x^2 + 4x + 4 + 3. Rewriting this, we have (x + 2)^2 + 3. So, the perfect square closest to the given expression is 2(x + 2)^2 + 3.