Expand the Trigonometric Expression (2x^2-3x-4)^2

${(2 x^{2} - 3 x - 4)}^{2}$

Step 1

Rewrite ${(2 x^{2} - 3 x - 4)}^{2}$ as $(2 x^{2} - 3 x - 4) (2 x^{2} - 3 x - 4)$ .

$(2 x^{2} - 3 x - 4) (2 x^{2} - 3 x - 4)$

Step 2

Expand $(2 x^{2} - 3 x - 4) (2 x^{2} - 3 x - 4)$ by multiplying each term in the first expression by each term in the second expression.

$2 x^{2} (2 x^{2}) + 2 x^{2} (- 3 x) + 2 x^{2} \cdot - 4 - 3 x (2 x^{2}) - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Step 3

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x^{2} x^{2} + 2 x^{2} (- 3 x) + 2 x^{2} \cdot - 4 - 3 x (2 x^{2}) - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$2 \cdot 2 (x^{2} x^{2}) + 2 x^{2} (- 3 x) + 2 x^{2} \cdot - 4 - 3 x (2 x^{2}) - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 \cdot 2 x^{2 + 2} + 2 x^{2} (- 3 x) + 2 x^{2} \cdot - 4 - 3 x (2 x^{2}) - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Add $2$ and $2$ .

$2 \cdot 2 x^{4} + 2 x^{2} (- 3 x) + 2 x^{2} \cdot - 4 - 3 x (2 x^{2}) - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Multiply $2$ by $2$ .

$4 x^{4} + 2 x^{2} (- 3 x) + 2 x^{2} \cdot - 4 - 3 x (2 x^{2}) - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Rewrite using the commutative property of multiplication.

$4 x^{4} + 2 \cdot - 3 x^{2} x + 2 x^{2} \cdot - 4 - 3 x (2 x^{2}) - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$4 x^{4} + 2 \cdot - 3 (x \cdot x^{2}) + 2 x^{2} \cdot - 4 - 3 x (2 x^{2}) - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$4 x^{4} + 2 \cdot - 3 (x^{1} x^{2}) + 2 x^{2} \cdot - 4 - 3 x (2 x^{2}) - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 x^{4} + 2 \cdot - 3 x^{1 + 2} + 2 x^{2} \cdot - 4 - 3 x (2 x^{2}) - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Add $1$ and $2$ .

$4 x^{4} + 2 \cdot - 3 x^{3} + 2 x^{2} \cdot - 4 - 3 x (2 x^{2}) - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Multiply $2$ by $- 3$ .

$4 x^{4} - 6 x^{3} + 2 x^{2} \cdot - 4 - 3 x (2 x^{2}) - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Multiply $- 4$ by $2$ .

$4 x^{4} - 6 x^{3} - 8 x^{2} - 3 x (2 x^{2}) - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Rewrite using the commutative property of multiplication.

$4 x^{4} - 6 x^{3} - 8 x^{2} - 3 \cdot 2 x \cdot x^{2} - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$4 x^{4} - 6 x^{3} - 8 x^{2} - 3 \cdot 2 (x^{2} x) - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$4 x^{4} - 6 x^{3} - 8 x^{2} - 3 \cdot 2 (x^{2} x^{1}) - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 x^{4} - 6 x^{3} - 8 x^{2} - 3 \cdot 2 x^{2 + 1} - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Add $2$ and $1$ .

$4 x^{4} - 6 x^{3} - 8 x^{2} - 3 \cdot 2 x^{3} - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Multiply $- 3$ by $2$ .

$4 x^{4} - 6 x^{3} - 8 x^{2} - 6 x^{3} - 3 x (- 3 x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Rewrite using the commutative property of multiplication.

$4 x^{4} - 6 x^{3} - 8 x^{2} - 6 x^{3} - 3 \cdot - 3 x \cdot x - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$4 x^{4} - 6 x^{3} - 8 x^{2} - 6 x^{3} - 3 \cdot - 3 (x \cdot x) - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Multiply $x$ by $x$ .

$4 x^{4} - 6 x^{3} - 8 x^{2} - 6 x^{3} - 3 \cdot - 3 x^{2} - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Multiply $- 3$ by $- 3$ .

$4 x^{4} - 6 x^{3} - 8 x^{2} - 6 x^{3} + 9 x^{2} - 3 x \cdot - 4 - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Multiply $- 4$ by $- 3$ .

$4 x^{4} - 6 x^{3} - 8 x^{2} - 6 x^{3} + 9 x^{2} + 12 x - 4 (2 x^{2}) - 4 (- 3 x) - 4 \cdot - 4$

Multiply $2$ by $- 4$ .

$4 x^{4} - 6 x^{3} - 8 x^{2} - 6 x^{3} + 9 x^{2} + 12 x - 8 x^{2} - 4 (- 3 x) - 4 \cdot - 4$

Multiply $- 3$ by $- 4$ .

$4 x^{4} - 6 x^{3} - 8 x^{2} - 6 x^{3} + 9 x^{2} + 12 x - 8 x^{2} + 12 x - 4 \cdot - 4$

Multiply $- 4$ by $- 4$ .

$4 x^{4} - 6 x^{3} - 8 x^{2} - 6 x^{3} + 9 x^{2} + 12 x - 8 x^{2} + 12 x + 16$

Simplify by adding terms.

Subtract $6 x^{3}$ from $- 6 x^{3}$ .

$4 x^{4} - 12 x^{3} - 8 x^{2} + 9 x^{2} + 12 x - 8 x^{2} + 12 x + 16$

Add $- 8 x^{2}$ and $9 x^{2}$ .

$4 x^{4} - 12 x^{3} + x^{2} + 12 x - 8 x^{2} + 12 x + 16$

Subtract $8 x^{2}$ from $x^{2}$ .

$4 x^{4} - 12 x^{3} - 7 x^{2} + 12 x + 12 x + 16$

Add $12 x$ and $12 x$ .

$4 x^{4} - 12 x^{3} - 7 x^{2} + 24 x + 16$

Expand the Trigonometric Expression (2x^2-3x-4)^2

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