Find the coefficients of $x^{2}$ and $x$ in $(x + 4) (x + 1) (x + 2)$

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Solution

The given binomials $(x + 4)$ , $(x + 1)$ and $(x + 2)$ .

Firstly multiply the first and second term as follows:

$(x + 4) (x + 1) (x + 2) = [(x \times x) + (x \times 1) + (4 \times x) + (4 \times 1)] (x + 2) = (x^{2} + x + 4 x + 4) (x + 2)$
$= (x^{2} + 5 x + 4) (x + 2)$

Now multiply the resulting expression:

$(x^{2} + 5 x + 4) (x + 2) = (x^{2} \times x) + (x^{2} \times 2) + (5 x \times x) + (5 x \times 2) + (4 \times x) + (4 \times 2)$
$= x^{3} + 2 x^{2} + 5 x^{2} + 10 x + 4 x + 8 = x^{3} + 7 x^{2} + 14 x + 8$

Therefore, $(x + 4) (x + 1) (x + 2) = x^{3} + 7 x^{2} + 14 x + 8$ .

Hence, the coefficient of $x^{2}$ is $7$ and coefficient of $x$ is $14$ .

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