Solve 6x^{7}(2x^{2}+5)^3+18x^9(2x^2+5)^2

$$6 x ^ { 7 } ( 2 x ^ { 2 } + 5 ) ^ { 3 } + 18 x ^ { 9 } ( 2 x ^ { 2 } + 5 ) ^ { 2 }$$

Evaluate

$$30\left(x^{2}+1\right)\left(2x^{2}+5\right)^{2}x^{7}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(2x^{2}+5\right)^{3}.

$$6x^{7}\left(8\left(x^{2}\right)^{3}+60\left(x^{2}\right)^{2}+150x^{2}+125\right)+18x^{9}\left(2x^{2}+5\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$6x^{7}\left(8x^{6}+60\left(x^{2}\right)^{2}+150x^{2}+125\right)+18x^{9}\left(2x^{2}+5\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$6x^{7}\left(8x^{6}+60x^{4}+150x^{2}+125\right)+18x^{9}\left(2x^{2}+5\right)^{2}$$

Use the distributive property to multiply 6x^{7} by 8x^{6}+60x^{4}+150x^{2}+125.

$$48x^{13}+360x^{11}+900x^{9}+750x^{7}+18x^{9}\left(2x^{2}+5\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x^{2}+5\right)^{2}.

$$48x^{13}+360x^{11}+900x^{9}+750x^{7}+18x^{9}\left(4\left(x^{2}\right)^{2}+20x^{2}+25\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$48x^{13}+360x^{11}+900x^{9}+750x^{7}+18x^{9}\left(4x^{4}+20x^{2}+25\right)$$

Use the distributive property to multiply 18x^{9} by 4x^{4}+20x^{2}+25.

$$48x^{13}+360x^{11}+900x^{9}+750x^{7}+72x^{13}+360x^{11}+450x^{9}$$

Combine 48x^{13} and 72x^{13} to get 120x^{13}.

$$120x^{13}+360x^{11}+900x^{9}+750x^{7}+360x^{11}+450x^{9}$$

Combine 360x^{11} and 360x^{11} to get 720x^{11}.

$$120x^{13}+720x^{11}+900x^{9}+750x^{7}+450x^{9}$$

Combine 900x^{9} and 450x^{9} to get 1350x^{9}.

$$120x^{13}+720x^{11}+1350x^{9}+750x^{7}$$

Expand

$$120x^{13}+720x^{11}+1350x^{9}+750x^{7}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(2x^{2}+5\right)^{3}.

$$6x^{7}\left(8\left(x^{2}\right)^{3}+60\left(x^{2}\right)^{2}+150x^{2}+125\right)+18x^{9}\left(2x^{2}+5\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$6x^{7}\left(8x^{6}+60\left(x^{2}\right)^{2}+150x^{2}+125\right)+18x^{9}\left(2x^{2}+5\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$6x^{7}\left(8x^{6}+60x^{4}+150x^{2}+125\right)+18x^{9}\left(2x^{2}+5\right)^{2}$$

Use the distributive property to multiply 6x^{7} by 8x^{6}+60x^{4}+150x^{2}+125.

$$48x^{13}+360x^{11}+900x^{9}+750x^{7}+18x^{9}\left(2x^{2}+5\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x^{2}+5\right)^{2}.

$$48x^{13}+360x^{11}+900x^{9}+750x^{7}+18x^{9}\left(4\left(x^{2}\right)^{2}+20x^{2}+25\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$48x^{13}+360x^{11}+900x^{9}+750x^{7}+18x^{9}\left(4x^{4}+20x^{2}+25\right)$$

Use the distributive property to multiply 18x^{9} by 4x^{4}+20x^{2}+25.

$$48x^{13}+360x^{11}+900x^{9}+750x^{7}+72x^{13}+360x^{11}+450x^{9}$$

Combine 48x^{13} and 72x^{13} to get 120x^{13}.

$$120x^{13}+360x^{11}+900x^{9}+750x^{7}+360x^{11}+450x^{9}$$

Combine 360x^{11} and 360x^{11} to get 720x^{11}.

$$120x^{13}+720x^{11}+900x^{9}+750x^{7}+450x^{9}$$

Combine 900x^{9} and 450x^{9} to get 1350x^{9}.

$$120x^{13}+720x^{11}+1350x^{9}+750x^{7}$$