1. **State the problem:** We are given the equation $$(ax + b)(ax^2 + cx + 1) = 25x^3 + 15x^2 - 28x + 7$$ where $a$, $b$, and $c$ are constants with $a > 0$. We need to find the values of $a$, $b$, and $c$. 2. **Expand the left side:** Use distributive property to expand: $$ (ax + b)(ax^2 + cx + 1) = ax \cdot ax^2 + ax \cdot cx + ax \cdot 1 + b \cdot ax^2 + b \cdot cx + b \cdot 1 $$ which simplifies to $$ a^2 x^3 + a c x^2 + a x + a b x^2 + b c x + b $$ 3. **Group like terms:** Group powers of $x$: $$ a^2 x^3 + (a c + a b) x^2 + (a + b c) x + b $$ 4. **Match coefficients with the right side:** The right side is $$ 25 x^3 + 15 x^2 - 28 x + 7 $$ Equate coefficients of corresponding powers of $x$: - For $x^3$: $$ a^2 = 25 $$ - For $x^2$: $$ a c + a b = 15 $$ - For $x^1$: $$ a + b c = -28 $$ - For constant term: $$ b = 7 $$ 5. **Solve for $a$:** Since $a^2 = 25$ and $a > 0$, we have $$ a = 5 $$ 6. **Substitute $a$ and $b$ into the $x^2$ coefficient equation:** $$ 5 c + 5 \times 7 = 15 $$ $$ 5 c + 35 = 15 $$ $$ 5 c = 15 - 35 $$ $$ 5 c = -20 $$ $$ c = \frac{\cancel{5} c}{\cancel{5}} = \frac{-20}{5} = -4 $$ 7. **Check the $x$ coefficient equation:** $$ a + b c = 5 + 7 \times (-4) = 5 - 28 = -23 $$ But the right side coefficient is $-28$, so re-check step 4 for the $x$ term. Actually, the $x$ term coefficient from expansion is: $$ a + b c $$ Substitute $a=5$, $b=7$, $c=-4$: $$ 5 + 7 \times (-4) = 5 - 28 = -23 $$ This does not match $-28$. Re-examine the expansion step 2 for the $x$ term: The $x$ terms come from: - $ax \cdot 1 = a x$ - $b \cdot c x = b c x$ Sum: $a x + b c x = (a + b c) x$ Given the mismatch, check if the original problem or coefficients are correct. Since the problem states the equation is true for all $x$, the coefficients must match exactly. Try to solve the system again with $b=7$: From $b=7$, $a=5$, and $a c + a b = 15$: $$ 5 c + 5 \times 7 = 15 \Rightarrow 5 c + 35 = 15 \Rightarrow 5 c = -20 \Rightarrow c = -4 $$ Check $a + b c$: $$ 5 + 7 \times (-4) = 5 - 28 = -23 $$ But the right side coefficient is $-28$, so the constant $b$ must be re-examined. 8. **Reconsider constant term:** From the constant term: $$ b = 7 $$ This is fixed. 9. **Re-express the $x^2$ coefficient equation:** $$ a c + a b = 15 $$ Substitute $a=5$: $$ 5 c + 5 b = 15 $$ $$ 5 c + 5 b = 15 $$ Divide both sides by 5: $$ c + b = 3 $$ 10. **Use $b=7$ from constant term:** $$ c + 7 = 3 \Rightarrow c = 3 - 7 = -4 $$ 11. **Check $x$ coefficient:** $$ a + b c = 5 + 7 \times (-4) = 5 - 28 = -23 $$ But the right side coefficient is $-28$, so this is inconsistent. 12. **Conclusion:** The system is inconsistent if $b=7$. Since $b$ is from the constant term, it must be $7$. Therefore, the $x$ coefficient on the right side should be $-23$ to match the expansion. Assuming the right side is correct, the only way to fix this is to re-check the original problem or accept the values: $$ a = 5, b = 7, c = -4 $$ **Final answer:** $$ a = 5, \quad b = 7, \quad c = -4 $$