1. Stating the problem: We need to find the determinant of the 3x3 matrix $$\begin{bmatrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{bmatrix}$$. 2. Recall the formula for the determinant of a 3x3 matrix $$\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$ is: $$\det = a(ei - fh) - b(di - fg) + c(dh - eg)$$ 3. Substitute the values from our matrix: $$a = x, b = 3, c = 7, d = 2, e = x, f = 2, g = 7, h = 6, i = x$$ 4. Calculate each part: - $$ei - fh = x \cdot x - 2 \cdot 6 = x^2 - 12$$ - $$di - fg = 2 \cdot x - 2 \cdot 7 = 2x - 14$$ - $$dh - eg = 2 \cdot 6 - x \cdot 7 = 12 - 7x$$ 5. Plug these back into the determinant formula: $$\det = x(x^2 - 12) - 3(2x - 14) + 7(12 - 7x)$$ 6. Expand each term: - $$x(x^2 - 12) = x^3 - 12x$$ - $$-3(2x - 14) = -6x + 42$$ - $$7(12 - 7x) = 84 - 49x$$ 7. Combine all terms: $$x^3 - 12x - 6x + 42 + 84 - 49x = x^3 - (12x + 6x + 49x) + (42 + 84) = x^3 - 67x + 126$$ 8. Final answer: $$\boxed{\det = x^3 - 67x + 126}$$