We are given the equation:

$y = (x + 3)^2 + (x + 4)^2$

First, let's expand each term on the right-hand side.

1. Expand $(x + 3)^2$: $(x + 3)^2 = x^2 + 6x + 9$
2. Expand $(x + 4)^2$: $(x + 4)^2 = x^2 + 8x + 16$

Now, add these two expanded terms together:

$y = (x^2 + 6x + 9) + (x^2 + 8x + 16)$

Simplifying:

$y = 2x^2 + 14x + 25$

Next, let's convert this equation into vertex form. Vertex form is:

$y = a(x - h)^2 + k$

We first factor out the 2 from the quadratic and linear terms:

$y = 2(x^2 + 7x) + 25$

Now, complete the square on $x^2 + 7x$. To complete the square, take half of 7 (which is $\frac{7}{2}$) and square it:

$\left(\frac{7}{2}\right)^2 = \frac{49}{4}$

Add and subtract $\frac{49}{4}$ inside the parentheses:

$y = 2\left(x^2 + 7x + \frac{49}{4} - \frac{49}{4}\right) + 25$

Simplify the expression inside the parentheses:

$y = 2\left(\left(x + \frac{7}{2}\right)^2 - \frac{49}{4}\right) + 25$

Now distribute the 2:

$y = 2\left(x + \frac{7}{2}\right)^2 - \frac{49}{2} + 25$

Convert 25 into a fraction:

$25 = \frac{50}{2}$

Finally, simplify:

$y = 2\left(x + \frac{7}{2}\right)^2 + \left(-\frac{49}{2} + \frac{50}{2}\right)$

$y = 2\left(x + \frac{7}{2}\right)^2 + \frac{1}{2}$

Thus, the equation in vertex form is:

$y = 2\left(x + \frac{7}{2}\right)^2 + \frac{1}{2}$

However, none of the answer choices match exactly. The closest match is:

$y = 2(x + 7/2)^2 - 1/2$

Would you like further details or have any questions?

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Related Questions:

1. How do you complete the square on a quadratic equation?
2. What is the significance of vertex form in quadratic functions?
3. How do you find the vertex of a quadratic equation in standard form?
4. Can you explain why we factor out the 2 when completing the square?
5. How does the transformation of $y = x^2$ change when moving to $y = a(x - h)^2 + k$?

Tip: Always double-check your arithmetic when completing the square, especially with fractions!