To simplify the expression $-2g(2a + 3) + 4f(1)$, let's break it down step by step.

Step 1: Define the functions

Given the functions:

• $f(x) = x^2 - 2x$
• $g(x) = 4x^2 - 6x + 8$

Step 2: Evaluate $f(1)$

We substitute $x = 1$ into the function $f(x)$:

$f(1) = 1^2 - 2(1) = 1 - 2 = -1$

Step 3: Evaluate $g(2a + 3)$

Now, substitute $x = 2a + 3$ into the function $g(x)$:

$g(2a + 3) = 4(2a + 3)^2 - 6(2a + 3) + 8$

First, expand $(2a + 3)^2$:

$(2a + 3)^2 = 4a^2 + 12a + 9$

Now substitute this into the expression for $g(2a + 3)$:

$g(2a + 3) = 4(4a^2 + 12a + 9) - 6(2a + 3) + 8$

Distribute the constants:

$g(2a + 3) = 16a^2 + 48a + 36 - 12a - 18 + 8$

Simplify the terms:

$g(2a + 3) = 16a^2 + 36a + 26$

Step 4: Substitute into the original expression

Now, substitute the values of $g(2a + 3)$ and $f(1)$ into the original expression:

$-2g(2a + 3) + 4f(1) = -2(16a^2 + 36a + 26) + 4(-1)$

Distribute the constants:

$= -32a^2 - 72a - 52 - 4$

Simplify:

$= -32a^2 - 72a - 56$

Final Simplified Expression:

$-32a^2 - 72a - 56$

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Would you like further details or have any questions about the process? Here are some related questions you can explore:

1. How would you evaluate $f(x)$ and $g(x)$ for different values of $x$?
2. How do you factor quadratic expressions like $-32a^2 - 72a - 56$?
3. What would happen if you evaluated $f(x)$ and $g(x)$ for non-integer values of $x$?
4. How can we find the vertex of the quadratic function $f(x)$?
5. What would the result be if we substituted $2a + 3$ into a different function $h(x)$?

Tip: When working with quadratic expressions, always check if there is an opportunity to factor or complete the square to simplify or analyze the function more deeply.