Algebra Examples

$g (x) = x^{2} (x + 4)$ at $x = t + 6$

Step 1

Replace the variable $x$ with $t + 6$ in the expression.

$g (t + 6) = {(t + 6)}^{2} ((t + 6) + 4)$

Step 2

Simplify the result.

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Step 2.1

Rewrite ${(t + 6)}^{2}$ as $(t + 6) (t + 6)$ .

$g (t + 6) = (t + 6) (t + 6) ((t + 6) + 4)$

Step 2.2

Expand $(t + 6) (t + 6)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$g (t + 6) = (t (t + 6) + 6 (t + 6)) ((t + 6) + 4)$

Step 2.2.2

Apply the distributive property.

$g (t + 6) = (t \cdot t + t \cdot 6 + 6 (t + 6)) ((t + 6) + 4)$

Step 2.2.3

Apply the distributive property.

$g (t + 6) = (t \cdot t + t \cdot 6 + 6 t + 6 \cdot 6) ((t + 6) + 4)$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $t$ by $t$ .

$g (t + 6) = (t^{2} + t \cdot 6 + 6 t + 6 \cdot 6) ((t + 6) + 4)$

Step 2.3.1.2

Move $6$ to the left of $t$ .

$g (t + 6) = (t^{2} + 6 \cdot t + 6 t + 6 \cdot 6) ((t + 6) + 4)$

Step 2.3.1.3

Multiply $6$ by $6$ .

$g (t + 6) = (t^{2} + 6 t + 6 t + 36) ((t + 6) + 4)$

Step 2.3.2

Add $6 t$ and $6 t$ .

$g (t + 6) = (t^{2} + 12 t + 36) ((t + 6) + 4)$

Step 2.4

Add $6$ and $4$ .

$g (t + 6) = (t^{2} + 12 t + 36) (t + 10)$

Step 2.5

Expand $(t^{2} + 12 t + 36) (t + 10)$ by multiplying each term in the first expression by each term in the second expression.

$g (t + 6) = t^{2} t + t^{2} \cdot 10 + 12 t \cdot t + 12 t \cdot 10 + 36 t + 36 \cdot 10$

Step 2.6

Simplify terms.

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Step 2.6.1

Simplify each term.

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Step 2.6.1.1

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

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Step 2.6.1.1.1

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

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Step 2.6.1.1.1.1

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

$g (t + 6) = t^{2} t + t^{2} \cdot 10 + 12 t \cdot t + 12 t \cdot 10 + 36 t + 36 \cdot 10$

Step 2.6.1.1.1.2

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

$g (t + 6) = t^{2 + 1} + t^{2} \cdot 10 + 12 t \cdot t + 12 t \cdot 10 + 36 t + 36 \cdot 10$

Step 2.6.1.1.2

Add $2$ and $1$ .

$g (t + 6) = t^{3} + t^{2} \cdot 10 + 12 t \cdot t + 12 t \cdot 10 + 36 t + 36 \cdot 10$

Step 2.6.1.2

Move $10$ to the left of $t^{2}$ .

$g (t + 6) = t^{3} + 10 \cdot t^{2} + 12 t \cdot t + 12 t \cdot 10 + 36 t + 36 \cdot 10$

Step 2.6.1.3

Multiply $t$ by $t$ by adding the exponents.

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Step 2.6.1.3.1

Move $t$ .

$g (t + 6) = t^{3} + 10 t^{2} + 12 (t \cdot t) + 12 t \cdot 10 + 36 t + 36 \cdot 10$

Step 2.6.1.3.2

Multiply $t$ by $t$ .

$g (t + 6) = t^{3} + 10 t^{2} + 12 t^{2} + 12 t \cdot 10 + 36 t + 36 \cdot 10$

Step 2.6.1.4

Multiply $10$ by $12$ .

$g (t + 6) = t^{3} + 10 t^{2} + 12 t^{2} + 120 t + 36 t + 36 \cdot 10$

Step 2.6.1.5

Multiply $36$ by $10$ .

$g (t + 6) = t^{3} + 10 t^{2} + 12 t^{2} + 120 t + 36 t + 360$

Step 2.6.2

Simplify by adding terms.

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Step 2.6.2.1

Add $10 t^{2}$ and $12 t^{2}$ .

$g (t + 6) = t^{3} + 22 t^{2} + 120 t + 36 t + 360$

Step 2.6.2.2

Add $120 t$ and $36 t$ .

$g (t + 6) = t^{3} + 22 t^{2} + 156 t + 360$

Step 2.7

The final answer is $t^{3} + 22 t^{2} + 156 t + 360$ .

$t^{3} + 22 t^{2} + 156 t + 360$

Step 3