We are given two equations:

1. $3x + 2y = 5$
2. $xy = -2$

We need to find the value of $27x^3 + 8y^3$.

Step-by-Step Solution:

Step 1: Recognize the form of the expression.

The expression $27x^3 + 8y^3$ can be rewritten as:

$27x^3 + 8y^3 = (3x)^3 + (2y)^3$

Step 2: Use the identity for sum of cubes.

We use the identity for the sum of cubes:

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Here, let $a = 3x$ and $b = 2y$. So we have:

$(3x)^3 + (2y)^3 = (3x + 2y)((3x)^2 - (3x)(2y) + (2y)^2)$

Step 3: Substitute $3x + 2y$ and simplify.

From the first equation, we know that $3x + 2y = 5$. Substituting this, we get:

$(3x)^3 + (2y)^3 = 5 \left( (3x)^2 - (3x)(2y) + (2y)^2 \right)$

Now, we calculate each term inside the parentheses:

• ( (3x)^2 =