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

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

Solve for y

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

Solution Steps

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

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

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

$$y=\left(4x^{2}+12x+9\right)\left(x-2\right)\left(x^{3}+12x^{2}+48x+64\right)$$

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

$$y=\left(4x^{3}+4x^{2}-15x-18\right)\left(x^{3}+12x^{2}+48x+64\right)$$

Use the distributive property to multiply 4x^{3}+4x^{2}-15x-18 by x^{3}+12x^{2}+48x+64 and combine like terms.

$$y=4x^{6}+52x^{5}+225x^{4}+250x^{3}-680x^{2}-1824x-1152$$