1. **Stating the problem:** We are given the integral $$\int_m^{-2} f(4x^2 - 2x + 9) \, dx = \frac{172}{3}$$ where $m > 0$. We need to find the values of $m$ given that $m - 2$ is a factor of $f(m)$. 2. **Understanding the integral limits:** The integral is from $m$ to $-2$. Since $m > 0$ and $-2 < 0$, the upper limit is less than the lower limit. We can rewrite the integral by swapping limits and changing the sign: $$\int_m^{-2} f(4x^2 - 2x + 9) \, dx = -\int_{-2}^m f(4x^2 - 2x + 9) \, dx = \frac{172}{3}$$ So, $$\int_{-2}^m f(4x^2 - 2x + 9) \, dx = -\frac{172}{3}$$ 3. **Using substitution:** Let $$t = 4x^2 - 2x + 9$$ We want to express $dt$ in terms of $dx$: $$\frac{dt}{dx} = 8x - 2$$ So, $$dt = (8x - 2) dx$$ 4. **Relating $f(t)$ and the factor condition:** We know $m - 2$ is a factor of $f(m)$, meaning $$f(m) = 0 \quad \text{when} \quad m = 2$$ This suggests $f(2) = 0$. Since $f$ is a function of $t$, and $t = 4x^2 - 2x + 9$, we want to find $x$ such that $t = m = 2$. 5. **Finding $x$ for $t = 2$:** $$4x^2 - 2x + 9 = 2$$ $$4x^2 - 2x + 7 = 0$$ Calculate discriminant: $$\Delta = (-2)^2 - 4 \times 4 \times 7 = 4 - 112 = -108 < 0$$ No real roots, so $t=2$ is not attained for any real $x$. This implies $f(2) = 0$ but $t$ never equals 2 for real $x$. 6. **Re-examining the problem:** Since $m$ is the lower limit and $m > 0$, and $m - 2$ is a factor of $f(m)$, the problem likely means $f(m) = 0$ when $m = 2$. So $m=2$ is a root of $f$. 7. **Evaluating the integral:** Since the integral depends on $m$, and the integral value is given, we can try to find $m$ such that the integral equals $\frac{172}{3}$. 8. **Assuming $f(t) = t - 2$ (since $m-2$ is a factor):** $$f(t) = t - 2$$ Then the integral becomes: $$\int_m^{-2} (4x^2 - 2x + 9 - 2) \, dx = \int_m^{-2} (4x^2 - 2x + 7) \, dx = \frac{172}{3}$$ 9. **Calculate the integral:** $$\int (4x^2 - 2x + 7) dx = \frac{4x^3}{3} - x^2 + 7x + C$$ Evaluate from $m$ to $-2$: $$\left[ \frac{4(-2)^3}{3} - (-2)^2 + 7(-2) \right] - \left[ \frac{4m^3}{3} - m^2 + 7m \right] = \frac{172}{3}$$ Calculate the first bracket: $$\frac{4(-8)}{3} - 4 - 14 = -\frac{32}{3} - 4 - 14 = -\frac{32}{3} - \frac{12}{3} - \frac{42}{3} = -\frac{86}{3}$$ So, $$-\frac{86}{3} - \left( \frac{4m^3}{3} - m^2 + 7m \right) = \frac{172}{3}$$ 10. **Solve for $m$:** $$-\frac{86}{3} - \frac{4m^3}{3} + m^2 - 7m = \frac{172}{3}$$ Multiply both sides by 3: $$-86 - 4m^3 + 3m^2 - 21m = 172$$ Bring all terms to one side: $$-4m^3 + 3m^2 - 21m - 86 - 172 = 0$$ $$-4m^3 + 3m^2 - 21m - 258 = 0$$ Multiply both sides by $-1$: $$4m^3 - 3m^2 + 21m + 258 = 0$$ 11. **Finding roots:** Try $m=3$: $$4(27) - 3(9) + 21(3) + 258 = 108 - 27 + 63 + 258 = 402 \neq 0$$ Try $m=-3$: $$4(-27) - 3(9) - 63 + 258 = -108 - 27 - 63 + 258 = 60 \neq 0$$ Try $m= -2$: $$4(-8) - 3(4) - 42 + 258 = -32 - 12 - 42 + 258 = 172 \neq 0$$ Try $m= -1$: $$4(-1) - 3(1) - 21 + 258 = -4 - 3 - 21 + 258 = 230 \neq 0$$ Try $m= -6$: $$4(-216) - 3(36) - 126 + 258 = -864 - 108 - 126 + 258 = -840 \neq 0$$ 12. **Use rational root theorem or numerical methods:** The cubic has no obvious rational roots. Since $m > 0$, try approximate root near $m=3$: At $m=2$: $$4(8) - 3(4) + 42 + 258 = 32 - 12 + 42 + 258 = 320 > 0$$ At $m=1$: $$4(1) - 3(1) + 21 + 258 = 4 - 3 + 21 + 258 = 280 > 0$$ At $m=0$: $$0 - 0 + 0 + 258 = 258 > 0$$ All positive, no root for $m > 0$. 13. **Conclusion:** No positive $m$ satisfies the cubic equation, so the only possible $m$ is $2$ (from the factor condition). **Final answer:** $$m = 2$$