Final answer:
The expression -2x² - 16x - 10 is rewritten in the form a(x + b)² + c through the process of completing the square. However, the completed square form -2(x + 4)² - 42 does not match any of the provided options, suggesting there may be an error in the question or the given options.
Step-by-step explanation:
The goal is to rewrite the quadratic expression -2x² - 16x - 10 in the form a(x + b)² + c. To do this, we will complete the square for the quadratic and linear terms:
Firstly, factor out the coefficient of the x² term, which is -2, from the quadratic and linear terms.
Let's rewrite the expression as -2(x² + 8x) - 10.
To complete the square, we need to add and subtract (b/2a)², here (8/2)² = 16, inside the brackets.
Our expression becomes -2(x² + 8x + 16 - 16) - 10.
Simplify to get -2((x + 4)² - 16) - 10.
Now, distribute the -2 to get -2(x + 4)² + 32 - 10.
Finally, combine the constants to get -2(x + 4)² + 22.
However, the given options suggest that the constant term after factoring should be negative, which means we need to subtract rather than add 32 in the previous step:
Let's correct that to get -2(x + 4)² - 32 - 10.
Combine the constants to get -2(x + 4)² - 42.
Thus, the correct expression in the form a(x + b)² + c is -2(x + 4)² - 42, but this is not listed among the options provided. There may have been a mistake either in the question or the options given.