Simplify P(x)=x^2(b^2x+a^2)+ab(x^5+1)

$P (x) = x^{2} (b^{2} x + a^{2}) + a b (x^{5} + 1)$

Step 1

Apply the distributive property.

$P (x) = x^{2} (b^{2} x) + x^{2} a^{2} + a b (x^{5} + 1)$

Step 2

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

Move $x$ .

$P (x) = x \cdot x^{2} b^{2} + x^{2} a^{2} + a b (x^{5} + 1)$

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$P (x) = x \cdot x^{2} b^{2} + x^{2} a^{2} + a b (x^{5} + 1)$

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

$P (x) = x^{1 + 2} b^{2} + x^{2} a^{2} + a b (x^{5} + 1)$

Add $1$ and $2$ .

$P (x) = x^{3} b^{2} + x^{2} a^{2} + a b (x^{5} + 1)$

Step 3

Apply the distributive property.

$P (x) = x^{3} b^{2} + x^{2} a^{2} + a b x^{5} + a b \cdot 1$

Step 4

Multiply $a$ by $1$ .

$P (x) = x^{3} b^{2} + x^{2} a^{2} + a b x^{5} + a b$