曲线 $y = \mathrm{f}(x)$ 的方程是一个三次多项式的形式。该曲线在 $(-1, 0)$ 处有一个极大值点，并经过坐标为 $(4, 25)$ 的点。曲线在 $x = 3$ 处的切线平行于方程为 $y = 24x + 5$ 的直线。求 $\mathrm{f}(x)$。

1. Set up the general equation for a cubic polynomial and find its derivative.

Let the cubic polynomial be $\mathrm{f}(x) = ax^3 + bx^2 + cx + d$.
The derivative is:
$\mathrm{f}'(x) = 3ax^2 + 2bx + c$

2. Use the coordinates of the maximum point to form two equations.

Since the curve has a maximum point at $(-1, 0)$, it passes through $(-1, 0)$ and its gradient there is $0$:
$\mathrm{f}(-1) = 0 \implies a(-1)^3 + b(-1)^2 + c(-1) + d = 0$
$-a + b - c + d = 0 \quad \text{--- (1)}$

$\mathrm{f}'(-1) = 0 \implies 3a(-1)^2 + 2b(-1) + c = 0$
$3a - 2b + c = 0 \quad \text{--- (2)}$

3. Use the other given point and the tangent gradient to form two more equations.

The curve passes through $(4, 25)$:
$\mathrm{f}(4) = 25 \implies a(4)^3 + b(4)^2 + c(4) + d = 25$
$64a + 16b + 4c + d = 25 \quad \text{--- (3)}$

The tangent at $x = 3$ is parallel to $y = 24x + 5$, so its gradient is $24$:
$\mathrm{f}'(3) = 24 \implies 3a(3)^2 + 2b(3) + c = 24$
$27a + 6b + c = 24 \quad \text{--- (4)}$

4. Solve the system of linear equations to find $a, b, c$ and $d$.

Subtracting (2) from (4) to eliminate $c$:
$(27a + 6b + c) - (3a - 2b + c) = 24 - 0$
$24a + 8b = 24 \implies 3a + b = 3 \implies b = 3 - 3a \quad \text{--- (5)}$

Substituting (5) into (2):
$3a - 2(3 - 3a) + c = 0 \implies c = 6 - 9a \quad \text{--- (6)}$

Subtracting (1) from (3) to eliminate $d$:
$(64a + 16b + 4c + d) - (-a + b - c + d) = 25 - 0$
$65a + 15b + 5c = 25 \implies 13a + 3b + c = 5 \quad \text{--- (7)}$

Substituting (5) and (6) into (7):
$13a + 3(3 - 3a) + (6 - 9a) = 5$
$13a + 9 - 9a + 6 - 9a = 5$
$-5a + 15 = 5 \implies -5a = -10 \implies a = 2$

Now find $b, c,$ and $d$:
$b = 3 - 3(2) = -3$
$c = 6 - 9(2) = -12$
From (1): $-2 + (-3) - (-12) + d = 0 \implies 7 + d = 0 \implies d = -7$

5. State the final cubic polynomial.

Substitute the values of $a, b, c,$ and $d$ back into the generic cubic polynomial:
$\mathrm{f}(x) = 2x^3 - 3x^2 - 12x - 7$