Solve a^{2}x+2ax+x+a^2y^2+2ay^2+y^2

$$a ^ { 2 } x + 2 a x + x + a ^ { 2 } y ^ { 2 } + 2 a y ^ { 2 } + y ^ { 2 }$$

Factor

$$\left(a+1\right)^{2}\left(x+y^{2}\right)$$

Solution Steps

Do the grouping a^{2}x+2ax+x+a^{2}y^{2}+2ay^{2}+y^{2}=\left(a^{2}x+2ax+x\right)+\left(a^{2}y^{2}+2ay^{2}+y^{2}\right), and factor out x in the first and y^{2} in the second group.

$$x\left(a^{2}+2a+1\right)+y^{2}\left(a^{2}+2a+1\right)$$

Factor out common term a^{2}+2a+1 by using distributive property.

$$\left(a^{2}+2a+1\right)\left(x+y^{2}\right)$$

Consider a^{2}+2a+1. Use the perfect square formula, p^{2}+2pq+q^{2}=\left(p+q\right)^{2}, where p=a and q=1.

$$\left(a+1\right)^{2}$$

Rewrite the complete factored expression.

$$\left(x+y^{2}\right)\left(a+1\right)^{2}$$

Evaluate

$$\left(a+1\right)^{2}\left(x+y^{2}\right)$$