Find the Product (3x^2+7)(6x^2-4x+5)

$(3 x^{2} + 7) (6 x^{2} - 4 x + 5)$

Expand $(3 x^{2} + 7) (6 x^{2} - 4 x + 5)$ by multiplying each term in the first expression by each term in the second expression.

$3 x^{2} (6 x^{2}) + 3 x^{2} (- 4 x) + 3 x^{2} \cdot 5 + 7 (6 x^{2}) + 7 (- 4 x) + 7 \cdot 5$

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$3 \cdot 6 x^{2} x^{2} + 3 x^{2} (- 4 x) + 3 x^{2} \cdot 5 + 7 (6 x^{2}) + 7 (- 4 x) + 7 \cdot 5$

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

Move $x^{2}$ .

$3 \cdot 6 (x^{2} x^{2}) + 3 x^{2} (- 4 x) + 3 x^{2} \cdot 5 + 7 (6 x^{2}) + 7 (- 4 x) + 7 \cdot 5$

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

$3 \cdot 6 x^{2 + 2} + 3 x^{2} (- 4 x) + 3 x^{2} \cdot 5 + 7 (6 x^{2}) + 7 (- 4 x) + 7 \cdot 5$

Add $2$ and $2$ .

$3 \cdot 6 x^{4} + 3 x^{2} (- 4 x) + 3 x^{2} \cdot 5 + 7 (6 x^{2}) + 7 (- 4 x) + 7 \cdot 5$

Multiply $3$ by $6$ .

$18 x^{4} + 3 x^{2} (- 4 x) + 3 x^{2} \cdot 5 + 7 (6 x^{2}) + 7 (- 4 x) + 7 \cdot 5$

Rewrite using the commutative property of multiplication.

$18 x^{4} + 3 \cdot - 4 x^{2} x + 3 x^{2} \cdot 5 + 7 (6 x^{2}) + 7 (- 4 x) + 7 \cdot 5$

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

Move $x$ .

$18 x^{4} + 3 \cdot - 4 (x \cdot x^{2}) + 3 x^{2} \cdot 5 + 7 (6 x^{2}) + 7 (- 4 x) + 7 \cdot 5$

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

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

$18 x^{4} + 3 \cdot - 4 (x^{1} x^{2}) + 3 x^{2} \cdot 5 + 7 (6 x^{2}) + 7 (- 4 x) + 7 \cdot 5$

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

$18 x^{4} + 3 \cdot - 4 x^{1 + 2} + 3 x^{2} \cdot 5 + 7 (6 x^{2}) + 7 (- 4 x) + 7 \cdot 5$

Add $1$ and $2$ .

$18 x^{4} + 3 \cdot - 4 x^{3} + 3 x^{2} \cdot 5 + 7 (6 x^{2}) + 7 (- 4 x) + 7 \cdot 5$

Multiply $3$ by $- 4$ .

$18 x^{4} - 12 x^{3} + 3 x^{2} \cdot 5 + 7 (6 x^{2}) + 7 (- 4 x) + 7 \cdot 5$

Multiply $5$ by $3$ .

$18 x^{4} - 12 x^{3} + 15 x^{2} + 7 (6 x^{2}) + 7 (- 4 x) + 7 \cdot 5$

Multiply $6$ by $7$ .

$18 x^{4} - 12 x^{3} + 15 x^{2} + 42 x^{2} + 7 (- 4 x) + 7 \cdot 5$

Multiply $- 4$ by $7$ .

$18 x^{4} - 12 x^{3} + 15 x^{2} + 42 x^{2} - 28 x + 7 \cdot 5$

Multiply $7$ by $5$ .

$18 x^{4} - 12 x^{3} + 15 x^{2} + 42 x^{2} - 28 x + 35$

Add $15 x^{2}$ and $42 x^{2}$ .

$18 x^{4} - 12 x^{3} + 57 x^{2} - 28 x + 35$