If $2a-b=3$ and $8a^3-b^3=999$, then find the value of $4a^2-b^2$.

The correct option is 63.

We are given two algebraic equations:
1) $2a-b=3$
2) $8a^3-b^3=999$

We need to find the value of:
$4a^2-b^2$

Let us analyze the terms using algebraic identities.
First, notice that $8a^3-b^3$ can be rewritten as ${\left(2a\right)}^3-b^3$.
We can use the algebraic identity for the difference of cubes:
$x^3-y^3=\left(x-y\right)\left(x^2+xy+y^2\right)$

By substituting $x=2a$ and $y=b$, we get:
$8a^3-b^3=\left(2a-b\right)\left(4a^2+2ab+b^2\right)$

Alternatively, we can express the difference of cubes in another standard form:
$x^3-y^3={\left(x-y\right)}^3+3xy\left(x-y\right)$

Substituting $x=2a$ and $y=b$ into this form gives:
$8a^3-b^3={\left(2a-b\right)}^3+3\left(2a\right)\left(b\right)\left(2a-b\right)$
$8a^3-b^3={\left(2a-b\right)}^3+6ab\left(2a-b\right)$

Now, substitute the known values ($2a-b=3$ and $8a^3-b^3=999$) into the equation:
$999={\left(3\right)}^3+6ab\left(3\right)$
$999=27+18ab$

Subtract 27 from both sides to isolate the term with $ab$:
$18ab=999-27$
$18ab=972$

Divide both sides by 18 to solve for $ab$:
$ab=\frac{972}{18}$
$ab=54$

Next, we want to find the value of $4a^2-b^2$. Using the difference of squares identity, we have:
$4a^2-b^2=\left(2a-b\right)\left(2a+b\right)$

We already know that $2a-b=3$. We need to find the value of $2a+b$.
We can use the algebraic relationship between the square of the sum and the square of the difference:
${\left(2a+b\right)}^2={\left(2a-b\right)}^2+4\left(2a\right)\left(b\right)$
${\left(2a+b\right)}^2={\left(2a-b\right)}^2+8ab$

Substitute the values of $2a-b=3$ and $ab=54$ into this relation:
${\left(2a+b\right)}^2=3^2+8\left(54\right)$
${\left(2a+b\right)}^2=9+432$
${\left(2a+b\right)}^2=441$

Taking the positive square root (since we look for a standard real solution that matches the positive multiple choice options):
$2a+b=\sqrt{441}=21$

Finally, calculate the value of the target expression:
$4a^2-b^2=\left(2a-b\right)\left(2a+b\right)$
$4a^2-b^2=3\times 21$
$4a^2-b^2=63$