Let's break down the problem step by step to help you solve the expressions.

First Expression:

We are given: $2x^2 + 4x - 5$ and the goal is to convert it into the form: $2(x+1)^2 - 7$

Start by factoring and simplifying:

1. The coefficient of $x^2$ is 2, so factor that out from the quadratic terms: $2(x^2 + 2x) - 5$
2. Complete the square inside the parentheses. To complete the square for $x^2 + 2x$, take half of 2 (which is 1), square it to get 1, and add and subtract this value: $2((x^2 + 2x + 1) - 1) - 5$
3. This simplifies to: $2((x+1)^2 - 1) - 5$
4. Distribute the 2: $2(x+1)^2 - 2 - 5$
5. Simplify the constants: $2(x+1)^2 - 7$ This matches the target expression, so the missing expression for the first blank is $2(x+1)^2 - 7$.

Second Expression:

We are given: $x^2 + 2x$ and the goal is to convert it into the form: $(x+1)^2 - 1$

Steps:

1. Complete the square for $x^2 + 2x$: $x^2 + 2x + 1 - 1$ $(x+1)^2 - 1$ This matches the target expression, so the missing expression for the second blank is $(x+1)^2 - 1$.

Third Expression:

We are given: $2x^2 + 8x - 2$ and the goal is to convert it into the form: $2(x+2)^2 - 10$

Steps:

1. Factor out the 2: $2(x^2 + 4x) - 2$
2. Complete the square for $x^2 + 4x$: $2((x^2 + 4x + 4) - 4) - 2$ $2((x+2)^2 - 4) - 2$
3. Distribute the 2: $2(x+2)^2 - 8 - 2$
4. Simplify the constants: $2(x+2)^2 - 10$ This matches the target expression, so the missing expression for the third blank is $2(x+2)^2 - 10$.

Would you like more details on these steps or have any questions?

Related Questions:

1. How do you complete the square for a quadratic expression?
2. Why do we factor out the leading coefficient when completing the square?
3. Can you complete the square for a quadratic expression with a negative coefficient?
4. How can completing the square help solve quadratic equations?
5. What other methods can be used to factor quadratic expressions?

Tip:

When completing the square, always ensure you adjust both sides of the equation if working with an equation, to maintain equality.