To rewrite the quadratic expression $-4x^2 - 16x - 19$ in vertex form, we can complete the square. The vertex form of a quadratic expression is given by:

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

where $(h, k)$ is the vertex of the parabola.

Steps:

1. Factor out the coefficient of $x^2$ from the first two terms (since the coefficient of $x^2$ is $-4$): $-4(x^2 + 4x) - 19$

2. Complete the square: To complete the square inside the parentheses, take half of the coefficient of $x$ (which is 4), square it, and add it inside the parentheses. Half of 4 is 2, and $2^2 = 4$. We must also subtract the extra 4 we just added by distributing the $-4$ outside the parentheses:

   $-4(x^2 + 4x + 4) - 19 + 16$

   (We added $4 \times -4 = -16$, so we need to add 16 outside the parentheses to balance it.)

3. Rewrite as a perfect square: Now the expression inside the parentheses is a perfect square: $-4(x + 2)^2 - 3$

So, the vertex form of $-4x^2 - 16x - 19$ is:

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

Here, the vertex is $(-2, -3)$.

Would you like further explanation or details?

Related Questions:

1. How do you find the axis of symmetry from a quadratic in vertex form?
2. What does the coefficient of $x^2$ tell you about the parabola's shape?
3. Can you convert the vertex form back to standard form?
4. How do you determine the direction of the parabola's opening?
5. What is the significance of completing the square in algebra?

Tip:

The vertex form is particularly useful because it gives you the vertex of the parabola directly, which is essential in graphing and understanding the parabola's behavior.