Solve y=(x+2)(2x+3)^2(3x+4)^3

$$y = ( x + 2 ) ( 2 x + 3 ) ^ { 2 } ( 3 x + 4 ) ^ { 3 }$$

Solve for y

$$y=\left(x+2\right)\left(2x+3\right)^{2}\left(3x+4\right)^{3}$$

Solution Steps

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

$$y=\left(x+2\right)\left(4x^{2}+12x+9\right)\left(3x+4\right)^{3}$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(3x+4\right)^{3}.

$$y=\left(x+2\right)\left(4x^{2}+12x+9\right)\left(27x^{3}+108x^{2}+144x+64\right)$$

Use the distributive property to multiply x+2 by 4x^{2}+12x+9 and combine like terms.

$$y=\left(4x^{3}+20x^{2}+33x+18\right)\left(27x^{3}+108x^{2}+144x+64\right)$$

Use the distributive property to multiply 4x^{3}+20x^{2}+33x+18 by 27x^{3}+108x^{2}+144x+64 and combine like terms.

$$y=108x^{6}+972x^{5}+3627x^{4}+7186x^{3}+7976x^{2}+4704x+1152$$