1. **State the problem:** We need to find the numbers to replace the boxes in the equation: $$\square x(x+3) + 9x(2x+1) = 5x(\square x + 6)$$ 2. **Rewrite the equation with variables for the unknown numbers:** Let the first box be $a$ and the second box be $b$. The equation becomes: $$a x (x+3) + 9x(2x+1) = 5x(b x + 6)$$ 3. **Expand each term:** - Left side: $$a x^2 + 3 a x + 18 x^2 + 9 x$$ - Right side: $$5 b x^2 + 30 x$$ 4. **Combine like terms on the left side:** $$ (a + 18) x^2 + (3 a + 9) x = 5 b x^2 + 30 x$$ 5. **Equate coefficients of like powers of $x$ on both sides:** - For $x^2$ terms: $$a + 18 = 5 b$$ - For $x$ terms: $$3 a + 9 = 30$$ 6. **Solve the $x$ coefficient equation for $a$:** $$3 a + 9 = 30 \implies 3 a = 21 \implies a = 7$$ 7. **Substitute $a=7$ into the $x^2$ coefficient equation to find $b$:** $$7 + 18 = 5 b \implies 25 = 5 b \implies b = 5$$ **Final answer:** The numbers to put in the boxes are $a=7$ and $b=5$. So the completed equation is: $$7 x (x+3) + 9 x (2x+1) = 5 x (5 x + 6)$$