Let 

$\begin{array}{r}\mathrm{\Delta }\left(x\right)=\left|\begin{array}{ccc}2x^3-3x^2& 5x+7& 2\\ 4x^3-7x& 3x+2& 1\\ 7x^3-8x^2& x-1& 3\end{array}\right|=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+a_4{x}^4\\ \end{array}$

To evaluate $a_i$ we differentiate $\mathrm{\Delta }\left(x\right) i$ times w.r.t. x and put $x=0$ or divide $\mathrm{\Delta }\left(x\right)$ by $x^4$ put $1/x=t$,

differentiate $(4-i)$ time w.r.t. t and put $t=0$ 

Problem Statement

Δ(x) = | 2x³ − 3x² 5x + 7 2 |
           | 4x³ − 7x 3x + 2 1 |
           | 7x³ − 8x² x − 1 3 |

= a₀ + a₁x + a₂x² + a₃x³ + a₄x⁴

Expansion (along the first row)

Δ(x) = (2x³ − 3x²) · | 3x + 2 1 ; x − 1 3 |
   − (5x + 7) · | 4x³ − 7x 1 ; 7x³ − 8x² 3 |
   + 2 · | 4x³ − 7x 3x + 2 ; 7x³ − 8x² x − 1 |

First minor

M₁ = (3x + 2)·3 − (x − 1)·1 = 8x + 7
T₁ = (2x³ − 3x²)(8x + 7) = 16x⁴ − 10x³ − 21x²

Second minor

M₂ = (4x³ − 7x)·3 − (7x³ − 8x²)·1 = 5x³ + 8x² − 21x
T₂ = −(5x + 7)(5x³ + 8x² − 21x) = −25x⁴ − 75x³ + 49x² + 147x

Third minor

M₃ = (4x³ − 7x)(x − 1) − (7x³ − 8x²)(3x + 2)
  = −17x⁴ + 6x³ + 9x² + 7x
T₃ = 2·M₃ = −34x⁴ + 12x³ + 18x² + 14x

Combine terms

Δ(x) = T₁ + T₂ + T₃
= (16x⁴ − 10x³ − 21x²) + (−25x⁴ − 75x³ + 49x² + 147x) + (−34x⁴ + 12x³ + 18x² + 14x)
= −43x⁴ − 73x³ + 46x² + 161x

Coefficients

a₀ = 0, a₁ = 161, a₂ = 46, a₃ = −73, a₄ = −43

If “Matril” has any extra constraints (e.g., no), tell me and I’ll switch the coefficients to a simple comma-separated line or a ``.