1. **State the problem:** Given the equations $a(x + y) = 7$ and $x - y = 3$, find the value of $8xy(x + y)x^2$. 2. **Analyze the given equations:** - From $a(x + y) = 7$, we can express $a = \frac{7}{x + y}$. - We are asked to find $8xy(x + y)x^2$ which depends on $x$, $y$, and $x + y$. 3. **Use the identity for $(x - y)^2$:** $$ (x - y)^2 = x^2 - 2xy + y^2 $$ Given $x - y = 3$, so: $$ 3^2 = x^2 - 2xy + y^2 = 9 $$ 4. **Express $x^2 + y^2$ in terms of $xy$:** We know: $$ (x + y)^2 = x^2 + 2xy + y^2 $$ So: $$ x^2 + y^2 = (x + y)^2 - 2xy $$ 5. **Relate $x^2 + y^2$ and $x^2 - 2xy + y^2$:** From step 3: $$ x^2 - 2xy + y^2 = 9 $$ From step 4: $$ x^2 + y^2 = (x + y)^2 - 2xy $$ 6. **Subtract the two expressions:** $$ (x^2 + y^2) - (x^2 - 2xy + y^2) = ((x + y)^2 - 2xy) - 9 $$ Simplify left side: $$ x^2 + y^2 - x^2 + 2xy - y^2 = 2xy $$ So: $$ 2xy = (x + y)^2 - 2xy - 9 $$ Rearranged: $$ 2xy + 2xy = (x + y)^2 - 9 $$ $$ 4xy = (x + y)^2 - 9 $$ 7. **Solve for $xy$:** $$ xy = \frac{(x + y)^2 - 9}{4} $$ 8. **Calculate $8xy(x + y)x^2$:** Rewrite as: $$ 8xy(x + y)x^2 = 8x^2 y (x + y) $$ 9. **Express $x^2 y$ in terms of $x$, $y$:** We can write: $$ 8xy(x + y)x^2 = 8x^2 y (x + y) = 8x (xy) (x + y) $$ 10. **We need values for $x$, $y$, and $x + y$ to proceed.** From $a(x + y) = 7$, $a$ is unknown, so let's find $x$ and $y$ from $x - y = 3$ and $xy$ from step 7. 11. **Let $S = x + y$, $P = xy$.** From step 7: $$ P = \frac{S^2 - 9}{4} $$ 12. **Use the quadratic equation for $x$ and $y$:** $$ t^2 - S t + P = 0 $$ 13. **Since $x - y = 3$, the difference of roots is:** $$ \sqrt{S^2 - 4P} = 3 $$ 14. **Substitute $P$ from step 11:** $$ \sqrt{S^2 - 4 \cdot \frac{S^2 - 9}{4}} = 3 $$ Simplify inside the square root: $$ \sqrt{S^2 - (S^2 - 9)} = 3 $$ $$ \sqrt{9} = 3 $$ This is true for any $S$. 15. **Therefore, $S = x + y$ is arbitrary, but from $a(x + y) = 7$, $a = \frac{7}{S}$.** 16. **Calculate $8xy(x + y)x^2$ in terms of $S$ and $x$:** Recall: $$ 8xy(x + y)x^2 = 8 P S x^2 $$ 17. **Express $x$ in terms of $S$ and $d=3$:** $$ x = \frac{S + d}{2} = \frac{S + 3}{2} $$ 18. **Substitute $P$ and $x$ into the expression:** $$ 8 P S x^2 = 8 \cdot \frac{S^2 - 9}{4} \cdot S \cdot \left(\frac{S + 3}{2}\right)^2 $$ 19. **Simplify step by step:** $$ = 8 \cdot \frac{S^2 - 9}{4} \cdot S \cdot \frac{(S + 3)^2}{4} $$ $$ = 8 \cdot \frac{S^2 - 9}{4} \cdot S \cdot \frac{(S + 3)^2}{4} $$ $$ = 8 \cdot \frac{S^2 - 9}{4} \cdot \frac{S (S + 3)^2}{4} $$ $$ = 8 \cdot \frac{S^2 - 9}{4} \cdot \frac{S (S + 3)^2}{4} $$ $$ = 8 \cdot \frac{S^2 - 9}{4} \cdot \frac{S (S + 3)^2}{4} $$ 20. **Multiply constants:** $$ 8 \times \frac{1}{4} \times \frac{1}{4} = 8 \times \frac{1}{16} = \frac{8}{16} = \frac{1}{2} $$ 21. **Final expression:** $$ \frac{1}{2} (S^2 - 9) S (S + 3)^2 $$ 22. **Since $S = x + y$ is unknown, the expression depends on $S$.** **Summary:** - The value of $8xy(x + y)x^2$ is $$\frac{1}{2} (x + y)^2 - 9) (x + y) (x + y + 3)^2$$. - Without a specific value for $x + y$, this is the simplified form. **Slug:** algebra expression **Subject:** algebra