If x − y = 4 and xy = 3, then determine the value of x³ − y³ + 5(x + y)².

The correct option is 240.

Given:
$x-y=4$
$xy=3$

Step 1: Calculate the value of $x^3-y^3$

We use the algebraic identity:
$x^3-y^3={(x-y)}^3+3xy(x-y)$

Substituting the given values into the identity:
$x^3-y^3=4^3+3\left(3\right)\left(4\right)$
$x^3-y^3=64+36$
$x^3-y^3=100$

Step 2: Calculate the value of ${(x+y)}^2$

We use the identity relating ${(x+y)}^2$ and ${(x-y)}^2$:
${(x+y)}^2={(x-y)}^2+4xy$

Substituting the given values:
${(x+y)}^2=4^2+4\left(3\right)$
${(x+y)}^2=16+12$
${(x+y)}^2=28$

Step 3: Determine the value of the target expression

We need to evaluate:
$x^3-y^3+5{(x+y)}^2$

Substituting the results from Step 1 and Step 2:
$x^3-y^3+5{(x+y)}^2=100+5\left(28\right)$
$x^3-y^3+5{(x+y)}^2=100+140$
$x^3-y^3+5{(x+y)}^2=240$

Thus, the final answer is 240.