MATL, 13 12 bytes

|stE:-GyZQ~)

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This uses the fact that, for integer coefficients, the absolute value of any root is strictly less than the sum of absolute values of the coefficients.

###Explanation

Consider input [1 5 6] as an example.

|    % Implicit input. Absolute value
     % STACK: [1 5 6]
s    % Sum
     % STACK: 12
t    % Duplicate
     % STACK: 12, 12
E    % Multiply by 2
     % STACK: 12, 24
:    % Range
     % STACK: 12, [1 2 ... 23 24]
-    % Subtract, elemet-wise
     % STACK: [11 10 ... -11 -12]
G    % Push input again
     % STACK: [11 10 ... -11 -12], [1 5 6]
y    % Duplicate from below
     % STACK: [11 10 ... -11 -12], [1 5 6], [11 10 ... -11 -12]
ZQ   % Polyval: values of polynomial at specified inputs
     % STACK: [11 10 ... -11 -12], [182 156 ... 72 90]
~    % Logical negation: turns nonzero into zero
     % STACK: [11 10 ... -11 -12], [0 0 ... 0] (contains 1 for roots)
)    % Index: uses second input as a mask for the first. Implicit display
     % STACK: [-3 -2]

MATL, 13 12 bytes

|stE:-GyZQ~)

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This uses the fact that, for integer coefficients, the absolute value of any root is strictly less than the sum of absolute values of the coefficients.

Explanation

Consider input [1 5 6] as an example.

|    % Implicit input. Absolute value
     % STACK: [1 5 6]
s    % Sum
     % STACK: 12
t    % Duplicate
     % STACK: 12, 12
E    % Multiply by 2
     % STACK: 12, 24
:    % Range
     % STACK: 12, [1 2 ... 23 24]
-    % Subtract, elemet-wise
     % STACK: [11 10 ... -11 -12]
G    % Push input again
     % STACK: [11 10 ... -11 -12], [1 5 6]
y    % Duplicate from below
     % STACK: [11 10 ... -11 -12], [1 5 6], [11 10 ... -11 -12]
ZQ   % Polyval: values of polynomial at specified inputs
     % STACK: [11 10 ... -11 -12], [182 156 ... 72 90]
~    % Logical negation: turns nonzero into zero
     % STACK: [11 10 ... -11 -12], [0 0 ... 0] (contains 1 for roots)
)    % Index: uses second input as a mask for the first. Implicit display
     % STACK: [-3 -2]

MATL, 13 bytes

|st_w&:GyZQ~)

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This uses the fact that, for integer coefficients, the absolute value of any root is bounded by the sum of absolute values of the coefficients.

###Explanation

Consider input [1 5 6] as an example.

|    % Implicit input. Absolute value
     % STACK: [1 5 6]
s    % Sum
     % STACK: 12
t    % Duplicate
     % STACK: 12, 12
_    % Arithmetical negation
     % STACK: 12, -12
w    % Swap
     % STACK: -12, 12
&:   % Binary range
     % STACK: [-12 -11 ... 12]
G    % Push input again
     % STACK: [-12 -11 ... 12], [1 5 6]
y    % Duplicate from below
     % STACK: [-12 -11 ... 12], [1 5 6], [-12 -11 ... 12]
ZQ   % Polyval: values of polynomial at specified inputs
     % STACK: [-12 -11 ... 12], [90 72 ... 210]
~    % Logical negation: turns nonzero into zero
     % STACK: [-12 -11 ... 12], [0 0 ... 0] (contains 1 for roots)
)    % Index: uses second input as a mask for the first. Implicit display
     % STACK: [-3 -2]

MATL, 13 12 bytes

|stE:-GyZQ~)

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This uses the fact that, for integer coefficients, the absolute value of any root is strictly less than the sum of absolute values of the coefficients.

###Explanation

Consider input [1 5 6] as an example.

|    % Implicit input. Absolute value
     % STACK: [1 5 6]
s    % Sum
     % STACK: 12
t    % Duplicate
     % STACK: 12, 12
E    % Multiply by 2
     % STACK: 12, 24
:    % Range
     % STACK: 12, [1 2 ... 23 24]
-    % Subtract, elemet-wise
     % STACK: [11 10 ... -11 -12]
G    % Push input again
     % STACK: [11 10 ... -11 -12], [1 5 6]
y    % Duplicate from below
     % STACK: [11 10 ... -11 -12], [1 5 6], [11 10 ... -11 -12]
ZQ   % Polyval: values of polynomial at specified inputs
     % STACK: [11 10 ... -11 -12], [182 156 ... 72 90]
~    % Logical negation: turns nonzero into zero
     % STACK: [11 10 ... -11 -12], [0 0 ... 0] (contains 1 for roots)
)    % Index: uses second input as a mask for the first. Implicit display
     % STACK: [-3 -2]

MATL, 13 bytes

|st_w&:GyZQ~)

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This uses the fact that, for integer coefficients, the absolute value of any root is bounded by the sum of absolute values of the coefficients.

###Explanation

Consider input [1 5 6] as an example.

|    % Implicit input. Absolute value
     % STACK: [1 5 6]
s    % Sum
     % STACK: 12
t    % Duplicate
     % STACK: 12, 12
_    % Arithemtical negation
     % STACK: 12, -12
w    % Swap
     % STACK: -12, 12
&:   % Binary range
     % STACK: [-12 -11 ... 12]
G    % Push input again
     % STACK: [-12 -11 ... 12], [1 5 6]
y    % Duplicate from below
     % STACK: [-12 -11 ... 12], [1 5 6], [-12 -11 ... 12]
ZQ   % Polyval: values of polynomial at specified inputs
     % STACK: [-12 -11 ... 12], [90 72 ... 210]
~    % Logical negation: turns nonzero into zero
     % STACK: [-12 -11 ... 12], [0 0 ... 0] (contains 1 for roots)
)    % Index: uses second input as a mask for the first. Implicit display
     % STACK: [-3 -2]

MATL, 13 bytes

|st_w&:GyZQ~)

Try it online!

This uses the fact that, for integer coefficients, the absolute value of any root is bounded by the sum of absolute values of the coefficients.

###Explanation

Consider input [1 5 6] as an example.

|    % Implicit input. Absolute value
     % STACK: [1 5 6]
s    % Sum
     % STACK: 12
t    % Duplicate
     % STACK: 12, 12
_    % Arithmetical negation
     % STACK: 12, -12
w    % Swap
     % STACK: -12, 12
&:   % Binary range
     % STACK: [-12 -11 ... 12]
G    % Push input again
     % STACK: [-12 -11 ... 12], [1 5 6]
y    % Duplicate from below
     % STACK: [-12 -11 ... 12], [1 5 6], [-12 -11 ... 12]
ZQ   % Polyval: values of polynomial at specified inputs
     % STACK: [-12 -11 ... 12], [90 72 ... 210]
~    % Logical negation: turns nonzero into zero
     % STACK: [-12 -11 ... 12], [0 0 ... 0] (contains 1 for roots)
)    % Index: uses second input as a mask for the first. Implicit display
     % STACK: [-3 -2]