1. **Problem statement:** Find the slope $m$ of the tangent to the curve $y = 7 + 5x^2 - 2x^3$ at $x = a$. 2. **Formula and rules:** The slope of the tangent line to a curve at a point is the derivative of the function evaluated at that point. 3. **Find the derivative:** $$y = 7 + 5x^2 - 2x^3$$ $$\frac{dy}{dx} = 0 + 10x - 6x^2 = 10x - 6x^2$$ 4. **Evaluate the slope at $x = a$:** $$m = 10a - 6a^2$$ 5. **Find equations of tangent lines at given points:** - At $(1, 10)$: - Slope: $m = 10(1) - 6(1)^2 = 10 - 6 = 4$ - Equation of tangent line using point-slope form: $$y - 10 = 4(x - 1)$$ $$y = 4x - 4 + 10 = 4x + 6$$ - At $(2, 11)$: - Slope: $m = 10(2) - 6(2)^2 = 20 - 24 = -4$ - Equation of tangent line: $$y - 11 = -4(x - 2)$$ $$y = -4x + 8 + 11 = -4x + 19$$ 6. **Summary:** - Slope at $x = a$ is $m = 10a - 6a^2$ - Tangent line at $(1,10)$: $y = 4x + 6$ - Tangent line at $(2,11)$: $y = -4x + 19$