Solve 3(4x+5)^2*4(5x+1)^2+(4x+5)*2(5+x)

$$3 ( 4 x + 5 ) ^ { 2 } \cdot 4 ( 5 x + 1 ) ^ { 2 } + ( 4 x + 5 ) \cdot 2 ( 5 + x )$$

Evaluate

$$4800x^{4}+13920x^{3}+12500x^{2}+3530x+350$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+5\right)^{2}.

$$3\left(16x^{2}+40x+25\right)\times 4\left(5x+1\right)^{2}+\left(4x+5\right)\times 2\left(5+x\right)$$

Multiply 3 and 4 to get 12.

$$12\left(16x^{2}+40x+25\right)\left(5x+1\right)^{2}+\left(4x+5\right)\times 2\left(5+x\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+1\right)^{2}.

$$12\left(16x^{2}+40x+25\right)\left(25x^{2}+10x+1\right)+\left(4x+5\right)\times 2\left(5+x\right)$$

Use the distributive property to multiply 12 by 16x^{2}+40x+25.

$$\left(192x^{2}+480x+300\right)\left(25x^{2}+10x+1\right)+\left(4x+5\right)\times 2\left(5+x\right)$$

Use the distributive property to multiply 192x^{2}+480x+300 by 25x^{2}+10x+1 and combine like terms.

$$4800x^{4}+13920x^{3}+12492x^{2}+3480x+300+\left(4x+5\right)\times 2\left(5+x\right)$$

Use the distributive property to multiply 4x+5 by 2.

$$4800x^{4}+13920x^{3}+12492x^{2}+3480x+300+\left(8x+10\right)\left(5+x\right)$$

Use the distributive property to multiply 8x+10 by 5+x and combine like terms.

$$4800x^{4}+13920x^{3}+12492x^{2}+3480x+300+50x+8x^{2}+50$$

Combine 3480x and 50x to get 3530x.

$$4800x^{4}+13920x^{3}+12492x^{2}+3530x+300+8x^{2}+50$$

Combine 12492x^{2} and 8x^{2} to get 12500x^{2}.

$$4800x^{4}+13920x^{3}+12500x^{2}+3530x+300+50$$

Add 300 and 50 to get 350.

$$4800x^{4}+13920x^{3}+12500x^{2}+3530x+350$$

Expand

$$4800x^{4}+13920x^{3}+12500x^{2}+3530x+350$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+5\right)^{2}.

$$3\left(16x^{2}+40x+25\right)\times 4\left(5x+1\right)^{2}+\left(4x+5\right)\times 2\left(5+x\right)$$

Multiply 3 and 4 to get 12.

$$12\left(16x^{2}+40x+25\right)\left(5x+1\right)^{2}+\left(4x+5\right)\times 2\left(5+x\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+1\right)^{2}.

$$12\left(16x^{2}+40x+25\right)\left(25x^{2}+10x+1\right)+\left(4x+5\right)\times 2\left(5+x\right)$$

Use the distributive property to multiply 12 by 16x^{2}+40x+25.

$$\left(192x^{2}+480x+300\right)\left(25x^{2}+10x+1\right)+\left(4x+5\right)\times 2\left(5+x\right)$$

Use the distributive property to multiply 192x^{2}+480x+300 by 25x^{2}+10x+1 and combine like terms.

$$4800x^{4}+13920x^{3}+12492x^{2}+3480x+300+\left(4x+5\right)\times 2\left(5+x\right)$$

Use the distributive property to multiply 4x+5 by 2.

$$4800x^{4}+13920x^{3}+12492x^{2}+3480x+300+\left(8x+10\right)\left(5+x\right)$$

Use the distributive property to multiply 8x+10 by 5+x and combine like terms.

$$4800x^{4}+13920x^{3}+12492x^{2}+3480x+300+50x+8x^{2}+50$$

Combine 3480x and 50x to get 3530x.

$$4800x^{4}+13920x^{3}+12492x^{2}+3530x+300+8x^{2}+50$$

Combine 12492x^{2} and 8x^{2} to get 12500x^{2}.

$$4800x^{4}+13920x^{3}+12500x^{2}+3530x+300+50$$

Add 300 and 50 to get 350.

$$4800x^{4}+13920x^{3}+12500x^{2}+3530x+350$$