Find the Product (a^2x+by^2)^2

${(a^{2} x + b y^{2})}^{2}$

Rewrite ${(a^{2} x + b y^{2})}^{2}$ as $(a^{2} x + b y^{2}) (a^{2} x + b y^{2})$ .

$(a^{2} x + b y^{2}) (a^{2} x + b y^{2})$

Expand $(a^{2} x + b y^{2}) (a^{2} x + b y^{2})$ using the FOIL Method.

Apply the distributive property.

$a^{2} x (a^{2} x + b y^{2}) + b y^{2} (a^{2} x + b y^{2})$

Apply the distributive property.

$a^{2} x (a^{2} x) + a^{2} x (b y^{2}) + b y^{2} (a^{2} x + b y^{2})$

Apply the distributive property.

$a^{2} x (a^{2} x) + a^{2} x (b y^{2}) + b y^{2} (a^{2} x) + b y^{2} (b y^{2})$

Simplify and combine like terms.

Simplify each term.

Multiply $a^{2}$ by $a^{2}$ by adding the exponents.

Move $a^{2}$ .

$a^{2} a^{2} x \cdot x + a^{2} x (b y^{2}) + b y^{2} (a^{2} x) + b y^{2} (b y^{2})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a^{2 + 2} x \cdot x + a^{2} x (b y^{2}) + b y^{2} (a^{2} x) + b y^{2} (b y^{2})$

Add $2$ and $2$ .

$a^{4} x \cdot x + a^{2} x (b y^{2}) + b y^{2} (a^{2} x) + b y^{2} (b y^{2})$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$a^{4} (x \cdot x) + a^{2} x (b y^{2}) + b y^{2} (a^{2} x) + b y^{2} (b y^{2})$

Multiply $x$ by $x$ .

$a^{4} x^{2} + a^{2} x (b y^{2}) + b y^{2} (a^{2} x) + b y^{2} (b y^{2})$

Multiply $b$ by $b$ by adding the exponents.

Move $b$ .

$a^{4} x^{2} + a^{2} x b y^{2} + b y^{2} a^{2} x + b \cdot b y^{2} y^{2}$

Multiply $b$ by $b$ .

$a^{4} x^{2} + a^{2} x b y^{2} + b y^{2} a^{2} x + b^{2} y^{2} y^{2}$

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

$a^{4} x^{2} + a^{2} x b y^{2} + b y^{2} a^{2} x + b^{2} (y^{2} y^{2})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a^{4} x^{2} + a^{2} x b y^{2} + b y^{2} a^{2} x + b^{2} y^{2 + 2}$

Add $2$ and $2$ .

$a^{4} x^{2} + a^{2} x b y^{2} + b y^{2} a^{2} x + b^{2} y^{4}$

Reorder $b$ and $a^{2}$ .

$a^{4} x^{2} + a^{2} b x y^{2} + a^{2} b x y^{2} + b^{2} y^{4}$

Add $a^{2} b x y^{2}$ and $a^{2} b x y^{2}$ .

$a^{4} x^{2} + 2 a^{2} b x y^{2} + b^{2} y^{4}$