Solve for t (4x^3+5)(2x^3+3)=8x^6+tx^3+15

$(4 x^{3} + 5) (2 x^{3} + 3) = 8 x^{6} + t x^{3} + 15$

Rewrite the equation as $8 x^{6} + t x^{3} + 15 = (4 x^{3} + 5) (2 x^{3} + 3)$ .

$8 x^{6} + t x^{3} + 15 = (4 x^{3} + 5) (2 x^{3} + 3)$

Simplify $(4 x^{3} + 5) (2 x^{3} + 3)$ .

Rewrite.

$8 x^{6} + t x^{3} + 15 = 0 + 0 + (4 x^{3} + 5) (2 x^{3} + 3)$

Simplify by adding zeros.

$8 x^{6} + t x^{3} + 15 = (4 x^{3} + 5) (2 x^{3} + 3)$

Expand $(4 x^{3} + 5) (2 x^{3} + 3)$ using the FOIL Method.

Apply the distributive property.

$8 x^{6} + t x^{3} + 15 = 4 x^{3} (2 x^{3} + 3) + 5 (2 x^{3} + 3)$

Apply the distributive property.

$8 x^{6} + t x^{3} + 15 = 4 x^{3} (2 x^{3}) + 4 x^{3} \cdot 3 + 5 (2 x^{3} + 3)$

Apply the distributive property.

$8 x^{6} + t x^{3} + 15 = 4 x^{3} (2 x^{3}) + 4 x^{3} \cdot 3 + 5 (2 x^{3}) + 5 \cdot 3$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$8 x^{6} + t x^{3} + 15 = 4 \cdot 2 x^{3} x^{3} + 4 x^{3} \cdot 3 + 5 (2 x^{3}) + 5 \cdot 3$

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$8 x^{6} + t x^{3} + 15 = 4 \cdot 2 (x^{3} x^{3}) + 4 x^{3} \cdot 3 + 5 (2 x^{3}) + 5 \cdot 3$

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

$8 x^{6} + t x^{3} + 15 = 4 \cdot 2 x^{3 + 3} + 4 x^{3} \cdot 3 + 5 (2 x^{3}) + 5 \cdot 3$

Add $3$ and $3$ .

$8 x^{6} + t x^{3} + 15 = 4 \cdot 2 x^{6} + 4 x^{3} \cdot 3 + 5 (2 x^{3}) + 5 \cdot 3$

Multiply $4$ by $2$ .

$8 x^{6} + t x^{3} + 15 = 8 x^{6} + 4 x^{3} \cdot 3 + 5 (2 x^{3}) + 5 \cdot 3$

Multiply $3$ by $4$ .

$8 x^{6} + t x^{3} + 15 = 8 x^{6} + 12 x^{3} + 5 (2 x^{3}) + 5 \cdot 3$

Multiply $2$ by $5$ .

$8 x^{6} + t x^{3} + 15 = 8 x^{6} + 12 x^{3} + 10 x^{3} + 5 \cdot 3$

Multiply $5$ by $3$ .

$8 x^{6} + t x^{3} + 15 = 8 x^{6} + 12 x^{3} + 10 x^{3} + 15$

Add $12 x^{3}$ and $10 x^{3}$ .

$8 x^{6} + t x^{3} + 15 = 8 x^{6} + 22 x^{3} + 15$

Move all terms not containing $t$ to the right side of the equation.

Subtract $8 x^{6}$ from both sides of the equation.

$t x^{3} + 15 = 8 x^{6} + 22 x^{3} + 15 - 8 x^{6}$

Subtract $15$ from both sides of the equation.

$t x^{3} = 8 x^{6} + 22 x^{3} + 15 - 8 x^{6} - 15$

Combine the opposite terms in $8 x^{6} + 22 x^{3} + 15 - 8 x^{6} - 15$ .

Subtract $8 x^{6}$ from $8 x^{6}$ .

$t x^{3} = 22 x^{3} + 15 + 0 - 15$

Add $22 x^{3} + 15$ and $0$ .

$t x^{3} = 22 x^{3} + 15 - 15$

Subtract $15$ from $15$ .

$t x^{3} = 22 x^{3} + 0$

Add $22 x^{3}$ and $0$ .

$t x^{3} = 22 x^{3}$

Divide each term in $t x^{3} = 22 x^{3}$ by $x^{3}$ and simplify.

Divide each term in $t x^{3} = 22 x^{3}$ by $x^{3}$ .

$\frac{t x^{3}}{x^{3}} = \frac{22 x^{3}}{x^{3}}$

Simplify the left side.

Cancel the common factor of $x^{3}$ .

Cancel the common factor.

$\frac{t x^{3}}{x^{3}} = \frac{22 x^{3}}{x^{3}}$

Divide $t$ by $1$ .

$t = \frac{22 x^{3}}{x^{3}}$

Simplify the right side.

Cancel the common factor of $x^{3}$ .

Cancel the common factor.

$t = \frac{22 x^{3}}{x^{3}}$

Divide $22$ by $1$ .

$t = 22$

Solve for t (4x^3+5)(2x^3+3)=8x^6+tx^3+15

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