1. **State the problem:** Solve the equation $ (4x-7)(x^2+2x) = 3 $ for $x$. 2. **Use the distributive property:** Expand the left side by multiplying each term in $4x-7$ by each term in $x^2+2x$: $$ (4x)(x^2) + (4x)(2x) - 7(x^2) - 7(2x) = 3 $$ 3. **Simplify each term:** $$ 4x^3 + 8x^2 - 7x^2 - 14x = 3 $$ 4. **Combine like terms:** $$ 4x^3 + (8x^2 - 7x^2) - 14x = 3 $$ $$ 4x^3 + x^2 - 14x = 3 $$ 5. **Bring all terms to one side to set equation to zero:** $$ 4x^3 + x^2 - 14x - 3 = 0 $$ 6. **Solve the cubic equation:** We look for rational roots using the Rational Root Theorem. Possible roots are factors of 3 over factors of 4: $\pm1, \pm\frac{1}{2}, \pm\frac{1}{4}, \pm3, \pm\frac{3}{2}, \pm\frac{3}{4}$. 7. **Test $x=1$:** $$4(1)^3 + 1^2 - 14(1) - 3 = 4 + 1 - 14 - 3 = -12 \neq 0$$ 8. **Test $x=3$:** $$4(3)^3 + 3^2 - 14(3) - 3 = 4(27) + 9 - 42 - 3 = 108 + 9 - 45 = 72 \neq 0$$ 9. **Test $x=-1$:** $$4(-1)^3 + (-1)^2 - 14(-1) - 3 = -4 + 1 + 14 - 3 = 8 \neq 0$$ 10. **Test $x=\frac{1}{2}$:** $$4\left(\frac{1}{2}\right)^3 + \left(\frac{1}{2}\right)^2 - 14\left(\frac{1}{2}\right) - 3 = 4\left(\frac{1}{8}\right) + \frac{1}{4} - 7 - 3 = \frac{1}{2} + \frac{1}{4} - 10 = -9.25 \neq 0$$ 11. **Test $x=\frac{3}{2}$:** $$4\left(\frac{3}{2}\right)^3 + \left(\frac{3}{2}\right)^2 - 14\left(\frac{3}{2}\right) - 3 = 4\left(\frac{27}{8}\right) + \frac{9}{4} - 21 - 3 = 13.5 + 2.25 - 24 = -8.25 \neq 0$$ 12. Since no rational roots found, use numerical methods or graphing to approximate roots. **Final answer:** The equation $4x^3 + x^2 - 14x - 3 = 0$ has no simple rational roots; solutions must be found using numerical methods.