We are given the function:

$f(x) = 2x^3 + Ax^2 + 7x - 7$

And we know that:

$f(2) = 7$

Substitute $x = 2$ into the function:

$f(2) = 2(2)^3 + A(2)^2 + 7(2) - 7$

Simplify the expression:

$f(2) = 2(8) + A(4) + 14 - 7$

$f(2) = 16 + 4A + 14 - 7$

$f(2) = 23 + 4A$

We are told that $f(2) = 7$, so:

$23 + 4A = 7$

Solve for $A$:

$4A = 7 - 23$

$4A = -16$

$A = \frac{-16}{4} = -4$

Thus, the value of $A$ is $-4$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you solve cubic equations in general?
2. What are other ways to find coefficients in a polynomial?
3. How would you approach solving a system of equations involving multiple variables?
4. What happens when a function’s value is undefined for a given $x$?
5. How do the coefficients of a polynomial affect its graph?

Tip: When solving equations, always simplify expressions step by step to avoid errors.