Calculus Examples

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

Step 1

Factor ${(2 x - 3)}^{2}$ out of ${(2 x - 3)}^{3} (x + 1) + (x - 3) {(2 x - 3)}^{2}$ .

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Step 1.1

Factor ${(2 x - 3)}^{2}$ out of ${(2 x - 3)}^{3} (x + 1)$ .

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

Step 1.2

Factor ${(2 x - 3)}^{2}$ out of $(x - 3) {(2 x - 3)}^{2}$ .

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

Step 1.3

Factor ${(2 x - 3)}^{2}$ out of ${(2 x - 3)}^{2} ((2 x - 3) (x + 1)) + {(2 x - 3)}^{2} (x - 3)$ .

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

Step 2

Expand $(2 x - 3) (x + 1)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

Move $x$ .

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

Step 3.1.1.2

Multiply $x$ by $x$ .

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

Step 3.1.2

Multiply $2$ by $1$ .

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

Step 3.1.3

Multiply $- 3$ by $1$ .

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

Step 3.2

Subtract $3 x$ from $2 x$ .

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

Step 4

Add $- x$ and $x$ .

${(2 x - 3)}^{2} (2 x^{2} + 0 - 3 - 3)$

Step 5

Add $2 x^{2}$ and $0$ .

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

Step 6

Subtract $3$ from $- 3$ .

${(2 x - 3)}^{2} (2 x^{2} - 6)$

Step 7

Factor.

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Step 7.1

Factor $2$ out of $2 x^{2} - 6$ .

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Step 7.1.1

Factor $2$ out of $2 x^{2}$ .

${(2 x - 3)}^{2} (2 (x^{2}) - 6)$

Step 7.1.2

Factor $2$ out of $- 6$ .

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

Step 7.1.3

Factor $2$ out of $2 x^{2} + 2 \cdot - 3$ .

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

Step 7.2

Remove unnecessary parentheses.

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