Answer :

To solve for [tex]\( y \)[/tex] in the equation

[tex]\[ \left|\begin{array}{ccc} y^2 & y & 1 \\ 8 & 4 & 10 \\ 9 & 3 & 6 \end{array}\right| = 60 \][/tex]

we need to follow these steps:

1. Set Up the Determinant Equation:
First, we write out the determinant of the given 3x3 matrix.

2. Calculate the Determinant:
The determinant of a 3x3 matrix

[tex]\[ \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} \][/tex]

is calculated as:

[tex]\[ a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \][/tex]

3. Substitute Values from the Matrix:
For our specific matrix, we substitute in the values:

[tex]\[ \left|\begin{array}{ccc} y^2 & y & 1 \\ 8 & 4 & 10 \\ 9 & 3 & 6 \end{array}\right| \][/tex]

So, using the formula:

[tex]\[ y^2 \left(4 \cdot 6 - 10 \cdot 3\right) - y\left(8 \cdot 6 - 10 \cdot 9\right) + 1 \left(8 \cdot 3 - 4 \cdot 9\right) \][/tex]

4. Simplify the Determinant Expression:
[tex]\[ y^2 (24 - 30) - y (48 - 90) + 1 (24 - 36) \][/tex]

[tex]\[ y^2 (-6) - y (-42) + (-12) \][/tex]

[tex]\[ -6y^2 + 42y - 12 \][/tex]

5. Set the Determinant Equal to 60:
The determinant is given to be equal to 60, so we set up the equation:

[tex]\[ -6y^2 + 42y - 12 = 60 \][/tex]

6. Rearrange and Solve the Equation:
Move 60 to the left side to set the equation to 0:

[tex]\[ -6y^2 + 42y - 12 - 60 = 0 \][/tex]

[tex]\[ -6y^2 + 42y - 72 = 0 \][/tex]

Divide by -6 to simplify:

[tex]\[ y^2 - 7y + 12 = 0 \][/tex]

7. Factor the Quadratic Equation:
Factor the quadratic equation:

[tex]\[ (y - 3)(y - 4) = 0 \][/tex]

8. Solve for [tex]\( y \)[/tex]:
Set each factor to zero and solve:

[tex]\[ y - 3 = 0 \quad \Rightarrow \quad y = 3 \][/tex]

[tex]\[ y - 4 = 0 \quad \Rightarrow \quad y = 4 \][/tex]

So, the solutions for [tex]\( y \)[/tex] are [tex]\( y = 3 \)[/tex] and [tex]\( y = 4 \)[/tex].

Thus, the correct answer is:
[tex]\[ \boxed{3, 4} \][/tex]