1. **State the problem:** Given the equations $a(x + y) = 7$ and $x - y = 3$, find the value of $8xy(x + y)$. 2. **Identify knowns and unknowns:** We know $a(x + y) = 7$ and $x - y = 3$. We want to find $8xy(x + y)$. 3. **Express $a$ in terms of $x$ and $y$:** From $a(x + y) = 7$, we have $$a = \frac{7}{x + y}$$ 4. **Use the identity for $(x + y)^2$:** Recall that $$ (x + y)^2 = x^2 + 2xy + y^2 $$ 5. **Use the identity for $(x - y)^2$:** Similarly, $$ (x - y)^2 = x^2 - 2xy + y^2 $$ 6. **Subtract the two equations:** $$ (x + y)^2 - (x - y)^2 = (x^2 + 2xy + y^2) - (x^2 - 2xy + y^2) = 4xy $$ 7. **Calculate $4xy$ using given values:** Given $x - y = 3$, so $$ (x - y)^2 = 3^2 = 9 $$ Let $S = x + y$, then $$ (x + y)^2 - 9 = 4xy $$ So, $$ 4xy = S^2 - 9 $$ 8. **Express $8xy(x + y)$ in terms of $S$ and $xy$:** $$ 8xy(x + y) = 2 \times 4xy \times S = 2(S^2 - 9)S = 2S^3 - 18S $$ 9. **Find $S = x + y$ using $a(x + y) = 7$:** We know $a = \frac{7}{S}$, but $a$ is unknown. We need to find $S$ another way. 10. **Use $a$ to find $xy$ or $S$:** Since $a$ is unknown and no other relation is given, we cannot find $S$ or $xy$ individually. 11. **Conclusion:** Without additional information about $a$, $x$, or $y$, or a relation involving $a$, we cannot find a numeric value for $8xy(x + y)$. **However, if we assume $a$ is a constant and $a(x + y) = 7$, then $S = \frac{7}{a}$.** Substitute $S = \frac{7}{a}$ into the expression: $$ 8xy(x + y) = 2S^3 - 18S = 2\left(\frac{7}{a}\right)^3 - 18\left(\frac{7}{a}\right) = 2\frac{343}{a^3} - \frac{126}{a} = \frac{686}{a^3} - \frac{126}{a} $$ This is the expression for $8xy(x + y)$ in terms of $a$. If $a$ is known, substitute to get the numeric value. **Final answer:** $$8xy(x + y) = \frac{686}{a^3} - \frac{126}{a}$$